hyperelliptic gromov-witten theory · we prove a “degree zero” version of this conjecture,...

HYPERELLIPTIC GROMOV-WITTEN THEORY

WILLIAM D. GILLAM

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy

in the Graduate School of Arts and Sciences

COLUMBIA UNIVERSITY

2008

ABSTRACT

HYPERELLIPTIC GROMOV-WITTEN THEORY

WILLIAM D. GILLAM

We study the genus zero orbifold Gromov-Witten theory of global quotients by invo-lutions. After reviewing the general theory of twisted stable maps, we explain how itsimplifies in this case. We explain when such a quotient admits a crepant resolution andgive an explicit construction of it by elementary methods, then we study the geometry ofthese resolutions. Next we study integrals of tautological classes on the moduli space ofhyperelliptic curves. Various Hurwitz-Hodge integrals are computed via localization andsolution of recursions, following the method of Cavalieri. The Crepant Resolution Con-jecture (CRC) of Bryan and Graber is discussed throughout. We prove a “degree zero”version of this conjecture, generalizing a result of Perroni. Later we study the local theoryof curves, where we consider a rank two vector bundle on a curve with the involution givenby multiplication by −1 on the fibers. We prove an equivariant version of the CRC incase the bundle is O ⊕O(−1) over P1. Finally, we prove the CRC for an involution of thevariety of complete flags in C3 taking a flag to its orthogonal complement with respect to anon-degenerate bilinear form on C3. This example has interesting connections to classicalgeometry.

Contents

Acknowledgements ii

1. Introduction 1

2. Twisted Curves and Stable Maps 2

3. Resolutions 14

4. Local Theory 18

5. The CRC in Degree Zero 33

6. Local Curves 40

7. Flags 60

Index of Notation 81

References 82

i

Acknowledgements

I am most indebted to my advisor, Michael Thaddeus, for suggesting several topicspursued in this thesis and for providing many important ideas throughout. I also wish tothank Renzo Cavalieri, Johan de Jong, and Davesh Maulik for answering various questionsand providing valuable suggestions. I am grateful to Dusa McDuff and Rahul Pandhari-pande for reading this thesis and serving on the defense committee. Though I cannotpinpoint his influence on this particular research, John Morgan has had a great impact onmy mathematical thinking over the years, and I wish to thank him as well. I also thank mycolleagues John Baldwin, Matt Deland, Joe Ross, and Dave Swinarski for participating indiscussions concerning this research and providing useful feedback.

ii

HYPERELLIPTIC GROMOV-WITTEN THEORY 1

1. Introduction

This thesis is about the genus zero orbifold Gromov-Witten (GW) theory of globalquotients of smooth projective varieties by involutions. Roughly, this is the study ofmaps from hyperelliptic curves to varieties exchanging the hyperelliptic involution witha specified involution of the target variety. In case the target is a point with trivial Z2

action, we end up studying a very nice compactification of the moduli space of hyperellipticcurves.

In Section 2 we recall the definitions of the relevant moduli spaces and explain somesimplifications we will use. We review the definition of Gromov-Witten invariants, thevirtual fundamental class, and the Chen-Ruan orbifold cohomology ring. We give thestatement of the Crepant Resolution Conjecture (CRC) of Bryan and Graber [14]. Roughlythe conjecture asserts that the GW theory of a Gorenstein orbifold should be equivalentto that of a crepant resolution of its coarse moduli space, should one exist. One of themain goals of this thesis is to work toward a proof of this conjecture.

Section 3 concerns the structure of crepant resolutions of quotients by involutions. Theseexist only when the fixed locus has codimension two. In this case, a crepant resolution isobtained by blowing up the fixed locus in the quotient, or first blowing up the fixed locusand then taking the quotient—it turns out that the order does not matter. We collectsome general facts about these resolutions for later use. We determine the normal bundleof the exceptional divisor in the crepant resolution.

In Section 4 we study integrals of tautological classes on the moduli space of hyperellipticcurves. We give a short, self-contained derivation of the necessary Hurwitz-Hodge integralsneeded in the rest of the thesis. Some of the results of this section were establishedpreviously by Faber and Pandharipande. Indeed, our approach follows Cavalieri and is notparticularly original. We did, however, perform the service of simplifying the computation,which had to be extracted from several different papers. Also in this section, we computemany other families of Hurwitz-Hodge integrals which arise naturally in Gromov-Wittentheory. We give a significant generalization of Cavalieri’s computations in the last part ofthis section to obtain a formula for integrals of classes of the form λgλg−iψ

k1 · · ·ψkm overthe moduli space of hyperelliptic curves.

The local theory developed in Section 4 is applied in Section 5 to prove the CRC “indegree zero” for global quotients X = [M/Z2]. It turns out that this can be done rathersimply using nothing more than basic algebraic topology and Chern class computations.For the cases we are interested in, this generalizes a result of Perroni [51], who proved theCRC for three point “degree zero” invariants in the case of orbifolds with transversal A1

or A2 singularities.

We continue studying the local theory in Section 6, where we consider a rank two vectorbundle on a curve with the involution given by multiplication by −1 on the fibers. Weprove an equivariant version of the CRC in case the bundle is O ⊕ O(−1) over P1.

Finally, in Section 7, we use the previous results to prove the CRC for an involution ofthe variety of complete flags in C3 taking a flag to its orthogonal complement with respectto a non-degenerate bilinear form on C3. This example has interesting connections toclassical geometry, which we also explain.

2 WILLIAM D. GILLAM

2. Twisted Curves and Stable Maps

Fix an algebraically closed field k of characteristic zero and let S be the category ofschemes over Spec k with the etale topology. It is understood that all principal bundles,gerbes, locally triviality and descent conditions, etc. are in the etale topology. Recall thata Deligne-Mumford (DM) stack is a stack X over S with representable, separated, andfinite diagonal which admits an etale covering by a scheme. Keel and Mori [42] provedthat every such stack X admits a coarse moduli space X → X which is a morphism (ofstacks) to an algebraic space X which is universal among such maps. This maps enjoysvarious other properties: it is proper, it gives a bijection on isomorphism classes of objectsover SpecK for any algebraically closed field K, etc. (All of this holds, more generally,for any algebraic stack with finite diagonal.) Any DM stack is, etale locally on its coarsemoduli space, a quotient stack of a scheme by a finite group (scheme). We use Zn todenote the cyclic group Z/nZ. We use the word “representable” to mean “representableby an algebraic space [8], [9]” throughout. There is no difference if we take our stacks overthe (opposite) category of k-algebras.

Fix a proper DM stack X , the target, whose coarse moduli space X is a projectivevariety. We wish to study the Gromov-Witten theory of X . It turns out that this iswell-behaved when X is an orbifold, that is, a smooth DM stack1 (meaning that it admitsan etale cover by a smooth scheme), even though its coarse moduli space X need not besmooth. We can define the appropriate twisted stable map spaces for any X as above,but the assumption that X is smooth is needed in order to define a perfect obstructiontheory and virtual fundamental class.

The naive thing to do is to simply study the moduli space of maps from curves to X .There is a perfectly good DM stack of such maps, but it is hardly ever proper, even ifX = BG for finite G. For example, if G = Z4

3, then the stack of Kontsevich stable mapsto X of genus 2 is not proper because a smooth curve C of genus 2 has connected principalG-bundles, but a nodal genus 2 curve of geometric genus 1 does not. The solution is toallow the domain curves to acquire stack structure along nodes.

2.1. Twisted stable maps. Fix a very ample sheaf O(1) on X. Let us recall, from [4]with influences from [47], the definition of the moduli space of stable maps from twistedcurves to X . Let µr denote the finite group scheme in S of rth roots of unity in k. Fix non-negative integers g, n and positive integers r1, . . . , rn. Consider the category Mg;n,r(X , d)whose objects are commutative diagrams of stacks over S with cartesian squares as below.

Gi

²²

// C

q

²²

f // X

²²B

=ÃÃA

AAAA

AAsi // C

π

²²

f // X

B

Note that there is one section si of π for each i = 1, . . . , n, though they are not simulta-neously displayed. We require these diagrams to satisfy the following properties.

1Here we do not require X to have generically trivial stabilizers. Indeed, X = BG will be of great interest.


(1) The base B is an object of S .(2) C is a DM stack and q displays C as its coarse moduli space, which may be an

algebraic space.(3) πq is a flat, proper morphism of DM stacks.(4) f is a representable morphism of DM stacks.(5) The bottom part of the diagram is a degree d genus g stable map, which means:

(a) π is flat and proper.(b) The sections si are disjoint and contained in the smooth locus of π.(c) The geometric fibers Cb of π are (reduced) connected curves with at worst

nodal singularities and with genus g (in the sense that h1(Cb,O) = g).(d) The map f has degree d on geometric fibers, with respect to our fixed O(1).(e) There are finitely many automorphisms.

(6) q is an isomorphism away from the nodes of C and the sections si.(7) For each i, Gi is a gerbe over B with fixed banding by µri

.(8) There is an etale neighborhood SpecA → C of any nodal geometric point of C

fitting into a diagram with cartesian square and etale maps as indicated

[SpecA[z, w]/〈zw − t, zn − x,wn − y〉 /µr] //

²²

C

²²SpecR[x, y]/〈xy − s〉

²²

SpecAetaleoo etale // C

²²SpecR ⊆

// B

with µr acting by ζ · x = ζx, ζ · y = ζ−1y.

Remark 2.1.1. Some of the flatness/properness conditions are redundant. The proper-ness in (5a) implies πq is proper since the coarse moduli space map is proper, and theflatness condition in (3) implies the one in (5a) by [4], 4.1.1. A priori, the geometricfibers Cb of condition (5c) are only algebraic spaces, but it is a standard fact that a onedimensional algebraic space is a scheme ([9], Theorem 2.7).

Morphisms in Mg;n,r(X , d) are the obvious 2-cartesian diagrams. In particular, thisis what we mean by “finitely many automorphisms” in condition (5e). Notice that theobjects themselves are categories, because the stacks C are categories, so Mg,n,r(X , d) isactually a 2-category. However, one can prove that there are no nonidentity 2-morphismsin this category, so it is equivalent to a category [4]. This is essentially because C mustgenerically be a scheme (or at least an algebraic space) because of condition (6).

The balancing condition (8) is necessary if the nodes of C can be smoothed. Since thepoint is to compactify the space of maps from smooth curves, it is reasonable to imposethis condition. This balancing condition is also needed to have a simple Riemann-Rochformula for the fiber curves Cb. The representability condition (4) is a sort of minimalitycondition meant to ensure that C cannot be “more stacky” than X . For example, ifX = X is a scheme, then C must be a scheme. The importance of this condition shouldbecome clear in the ensuing discussion.

Theorem 2.1.1. ([7], Theorem 1.4.1) Mg;n,r(X , d) is a proper DM stack over S withprojective coarse moduli space.

4 WILLIAM D. GILLAM

In [7], this theorem is proved by using a general criterion of Artin for a stack to bealgebraic, and hence the proof involves verifying a long list of technical conditions. Theproblem was that, at the time, there was nothing like a moduli space of twisted curves,or a Hilbert scheme of maps from a twisted curve to X , which would have been anobvious starting point. However, M. Olsson [47] later gave an alternative construction ofthe moduli stack of twisted stable curves by relating them to curves with log structures.Olsson also constructed an algebraic Hom stack for very general stacks in [48] and usedthis to give an alternative construction of Mg;n,r(X , d) which is more illuminating thanthe original one. See [1] for further commentary.

The purpose of the rest of this section is to explain how Mg;n,r(X , d) simplifies ifX = [M/G] is a global quotient stack, and then to discuss the evaluation maps in thisparticular case.

By definition, an object of Mg;n,r([M/G], d) is a diagram with cartesian squares asbelow.

Ci

²²

// C

p

²²

f // M

²²Gi

²²

// C

q

²²

f // [M/G]

²²B

=ÂÂ@

@@@@

@@@

si // C

π

²²

f // M/G

B

Here p : C → C is a principal G-bundle and g is a G-equivariant map. Notice that f mustbe representable, so C must be in S , or at least be an algebraic space over S becauseC is obtained by base change of f along M → [M/G] and M is a scheme. In particular,the µri

-gerbe Gi can be expressed as a global quotient [Ci/G] of an algebraic space by G,which places serious restrictions on its structure. For example, if we look over a point bof B, then Gi|b should be isomorphic to Bµri

, but it is simultaneously a global quotientof Ci|b by G, so Ci|b must consist of finitely many points, acted on transitively by G (elsethe coarse moduli space Ci/G would not be the single point b) so that the stabilizers ofthese points (which are conjugate subgroups of G) are isomorphic to µri

. In particular,ri must divide |G| and if ri = |G|, we must have G ∼= µri

and the G-action on Ci must betrivial, hence we must have Ci = B and the µri

-gerbe Gi is displayed as the trivial gerbe[B/G] ∼= B ×Bµri

.

For simplicity, we now restrict to the case where G is cyclic of prime order p. Then itfollows from the above discussion that Mg;n,r([M/G], d) is empty unless r of the ri areequal to p, and the other u := n − r of the ri are 1, for some nonnegative integers r, u.Then we may as well forget the stack C and define an object of Mg;r,u([M/G], d) to be a


diagram with cartesian square as below.

C

p

²²

f // M

²²C

f //

π

²²

M/G

B

Ri

99

Ui

II

Here the Ui are sections of π for i = 1, . . . , u and the Ri are sections of πp for i = 1, . . . , r.We require the following.

(1) (C, pR1, . . . , pRr, U1, . . . , Uu, π, f) is a degree d genus g stable map.(2) (C, R1, . . . , Rr, πp) is a prestable curve.(3) C is equipped with a G-action making g a G-equivariant map with C equal to the

coarse quotient C/G.(4) The G-action on C makes p a principal G-bundle away from the Ri and nodes of

C. The G-action is trivial on the Ri.(5) Balancing. Near nodal geometric points of C, there are diagrams as in condition

(8) above, except we do not take the stack quotient and we replace C with C.

This data is equivalent to the data in the previous definition because the stack C canbe recovered as the quotient stack [C/G]. Note that we recover the original condition (3)by demanding the same conditions of the etale cover C → [C/G]. We recover the trivialµp-gerbe Gi as the stack quotient of Ri by the trivial G-action.

We view the above left part of the above diagram as giving a family C of Z2 coversof genus g curves C parameterized by B. The sections Ri, Ui are viewed as determiningmarked points in the smooth loci of the curves in this family. The right part of the abovediagram is the data of the stable map.

2.2. Evaluation maps. Assuming that G is cyclic of order p we now fix isomorphismsµp

∼= G ∼= Zp once and for all. Set G∗ := G \ IdG. Observe that for each i = 1, . . . , uthere is a morphism of stacks over S

ei : Mg;r,u([M/G], d) → [M/G]

given by taking a diagram as above to the principal G-bundle over B with G-equivariantmap to M obtained by base changing C along Ui. For i = 1, . . . , r, we can consider apoint b in B and the G = Zp action on TRi(b)Cb. Then the generator 1 ∈ G = Zp acts bysome root of unity and comparing this with the image of 1 under our fixed identificationZp = µp, we get a locally constant function m : B → G∗. We use this to define anevaluation morphism

ei : Mg;r,u([M/G], d) →∐

g∈G∗MG

given by base changing along Ri and using m to decide which copy of MG to map to.(Note that the maps to B → M obtained by base change along Ri factor through theG-fixed locus MG because g is G-equivariant and the G action on Ri is trivial.)

6 WILLIAM D. GILLAM

The stackMg;r,u([M/G], d) can be divided into components by considering preimages ofcomponents of the inertia stack under the evaluation morphisms. Many such componentswill be empty for monodromy reasons. For example, M0;r,u([M/Z2], d) is empty unless r iseven because the map C → C in the above diagram must be ramified at an even numberof points (over each geometric point of B) by Riemann-Hurwitz. Since we are mostlyinterested in Z2 global quotients, we will not introduce extra notation for componentswhere we impose more elaborate monodromy conditions.

Remark 2.2.1. The stack

IX = [M/G]∐ ∐

g∈G∗MG

is nothing but the rigidified cyclotomic inertia stack considered in [4] (see Section 4.2 in[1]). Indeed, using our identifications µp = G = Zp, this is the disjoint union, over all r,of the stacks of representable maps from µr gerbes to [M/G], modulo the action of thegroup stack Bµp. (There are no such representable maps unless r = 0 or r = p.) This isa rigidification of the inertia stack IX defined as the fibered product over the diagonalmaps IX := X ×X ×X X .

2.3. Gromov-Witten theory. Working over k = C, a finite type DM stack over S givesrise to a stack in the category of topological spaces by choosing an etale presentation R ⇒U of X and considering the stack determined by the underlying presentation of topologicalspaces. One can then assign cohomology groupsH i(X ) as in Behrend’s lectures [11]. Withrational coefficients, these are nothing but the cohomology groups of the course modulispace X. When X = M/G is a global quotient (topological space) by a finite group, atheorem of Grothendieck implies that the quotient map M → X induces an isomorphismfrom H∗(X;Q) onto the G-invariant cohomology H∗(M ;Q)G of M . The moduli spaceMg;n,r(X , d) splits into components indexed by d ∈ H2(X;Z), so from now on we willuse d to denote an effective homology class d ∈ Heff

2 (X;Z).

The stack of twisted stable maps M = Mg;n,r(X , d) carries a virtual fundamental class[M]vir in the Chow group A∗(M) which is used to define Gromov-Witten invariants.2

We will give a cursory treatment of the construction of the virtual fundamental class. LetMtw = Mtw

g;n,r be the Artin stack of (not necessarily stable) twisted curves (Theorem 1.9of Olsson’s paper [47] asserts that Mtw is a smooth Artin stack and that the forgetfulmap Mtw → Mg,n taking a twisted curve to its coarse moduli space is relatively of DMtype) and let Lτ be the cotangent complex (c.f. [41], [44], [49], and especially [50]) of theforgetful map τ : M→ Mtw which retains only the domain curve.

2There is a well-developed theory of Chow groups for DM (even Artin) stacks [54], [43], but for our purposes,rational coefficients suffice and in this case, the Chow group A∗(X )Q of a DM stack X is nothing but theChow group A∗(X)Q of its coarse moduli space. One only needs to weight by the size of automorphismgroups: If Y is a closed integral substack of X whose generic point has isotropy group of order n, then[Y ] = 1

n[Y ] under the identification A∗(X ) = A∗(X). For example, BZn → Spec k has degree 1

n. The

usual proper pushforward, flat pullback, and refined Gysin homomorphisms as in Fulton’s book [30] arealso defined for DM stacks.


Referring to the universal diagram

Cf−−−−→ X

π

yM

we set B• equal to (Rπ∗f∗TX )∨ in the derived category of coherent sheaves onM. Whenthe target X is smooth, there is a morphism

φ : B• → Lτ

in the derived category of M which is a perfect (relative) obstruction theory. This meansthat B• is locally of perfect amplitude [−1, 0], that is, it is locally quasi-isomorphic to acomplex B−1 → B0 of finite rank locally free sheaves supported in degrees −1, 0 withh0(φ) an isomorphism and h−1(φ) surjective. J. Wise explained the construction of thismap to me. Consider the diagram

Mτ

²²

Cπoo

p

²²

f // X

²²Mtw Coo // SpecC

where the left square is cartesian, but not necessarily the right. Here C → Mtw is theuniversal twisted curve. By general theory of the cotangent complex ([41] Corollaire 2.2.3,1.2.7.2), we have an isomorphism π∗Lτ

∼= Lp (using flatness of π) and a natural map

f∗LX → Lp∼= π∗Lτ .

Since X is smooth, we have LX∼= ΩX so we may take the dual of this map to get a map

π∗L∨τ → f∗TX .

Since π∗ is left adjoint to Rπ∗, this is the same thing as a morphism

L∨τ → Rπ∗f∗TX .

Dualizing again and using the fact the the cotangent complex is (or rather “can be rep-resented by”) a complex of locally frees (by construction) hence isomorphic to its doubledual, we get the desired map φ. The fact that φ is a perfect (relative) obstruction theoryis proved in [13] Proposition 6.2.

The intrinsic normal cone machinery of Behrend and Fantechi [13] then produces avirtual fundamental class

[M]vir = [M, B•]vir ∈ Ae(M),

where the expected dimension e is given as the difference

e = rankB0 − rankB−1.

Note that one must use the results of Kresch ([43], Section 6.2) to produce this class with-out some additional assumptions as in the remark below. This virtual class is obtained byembedding the intrinsic normal cone Cτ of τ in the vector bundle stack B = (h1/h0)((B•)∨)and intersecting with the zero section of B:

[M, B•]vir = 0!B[Cτ ].

8 WILLIAM D. GILLAM

Since these cone stacks and vector bundle stacks are not of DM type, the results of Kreschare needed to define this. The expected dimension e of the stable map spaces can becalculated by Riemann-Roch for stacky curves ([4], Section 7.2).

Remark 2.3.1. Suppose the target stack X is a global quotient [M/G] of a quasi-projective scheme M by a finite group G. Then the main result of [6] is that the stacks ofstable maps Mg,n(X , β) have the resolution property : every coherent sheaf is the quotientof a vector bundle. Under the same hypothesis on X , the same paper also shows thatthe stable map space Mg,n(X , β) admits a closed embedding in a smooth DM stack.Assuming the resolution property, Lemma 2.5 of [12] asserts that any perfect obstructiontheory can be written globally as a two term complex of vector bundles E−1 → E0. Thelatter properties are necessary for the virtual localization machinery as developed in [34],which we use throughout.

Remark 2.3.2. As far as I can tell, one can prove directly that the complex (Rπ∗f∗TX )∨defining the perfect obstruction theory is of perfect amplitude [−1, 0] by following the proofof Proposition 5 in [11], using Theorem 1.13 of [47] to reduce to the case of stable curves.

Under the assumptions in Remark 2.3.1 (which hold for any stable map space consideredhere), the virtual fundamental class can be constructed concretely as follows. Choose aglobal resolution: a quasi-isomorphism

[B−1 → B0] ∼= (Rπ∗f∗TX )∨

with the Bi locally free (of finite rank). Throughout this discussion, we will write Bi forthe vector bundle

Spec SymBi

whose sheaf of sections is H om(Bi,OM). Form the cartesian square below.

C(B•) Â Ä //

²²

B1

²²Cτ

Â Ä // [B1/B0]

The class 0!B1

[C(B•)] is the virtual fundamental class [M, B•]vir and is, in particular,independent of the choice of global resolution ([13], Proposition 5.3).

It is possible to be even more explicit about the intrinsic normal cone and the re-lationship between the relative and absolute obstruction theories (we continue with theassumptions in Remark 2.3.1). The obstruction theory map φ : B• → Lτ can be completedto a commutative diagram of distinguished triangles

A•

²²

// E•

²²

// B•

φ

²²

// A•[1]

²²τ∗LMtw // LM // Lτ

// τ∗LMtw [1]

where A• → τ∗LMtw is a quasi-isomorphism and E• → LM is itself a perfect obtructiontheory (see Appendix B of [34]). Furthermore, since Mtw is smooth, the two virtualclasses [M, E•]vir, and [M, B•]vir agree by Proposition A.1 of [15]. The first of thesevirtual classes is defined by intersecting the intrinsic normal cone of M with the zerosection of E = (h1/h0)((E•)∨), which is described in [14] as follows.


Fix a closed embedding M → S to a smooth DM stack S; let I be the ideal sheaf ofM in S. Let

NM/S = Spec Sym(I/I2), CM/S = Spec⊕nIn/In+1

be the normal sheaf and cone of M in S, respectively. Only the truncation τ≤−1LM isneeded in the construction. By general theory of the cotangent complex ([41], Corol-laire 3.1.3), we have a quasi-isomorphism

τ≤−1LM ∼= [I/I2 → ΩS ]

(the right complex is in degrees −1, 0) and, using the hypotheses in the remark, we mayassume our obstruction theory is given by a morphism of complexes

φ : [E−1 → E0] → [I/I2 → ΩS ]

with E−1, E0 locally free. The cohomology hypotheses on φ imply that the sequence

E−1 → E0 ⊕ I/I2 → ΩS → 0

is exact. Let Q be the kernel of E0⊕I/I2 → ΩS so exactness in the middle gives a naturalsurjection E−1 ³ Q, and taking Spec Sym we get a closed embedding C(Q) → E1. Thevector bundle

TS = Spec SymΩS

acts on E0 ×M NM/S via the right map of the above SES and preserves the subcone

E0 ×M CM/S → E0 ×M NM/S .

(That is, CM/S is a TS-cone in the language of [13]. See Lemma 3.2 in [13].) The quotientstack

C ′ := [(E0 ×M CM/S)/TS]

embeds in C(Q) and hence in E1 and the virtual fundamental class is obtained by inter-secting with the zero section of E1:

[M, E•]vir = 0!E1

[C ′].

It is perfectly reasonable to take the virtual class as a black box, since one typicallyonly needs to make use of various formal properties it satisfies. We refer to [23], [4], [16]for lists of such properties. In case M is smooth (and hence CM is a trivial cone), and(Rπ∗f∗TX )∨ is globally isomorphic to a locally free sheaf E−1 in degree −1, the conestack [E1/E0] is just the vector bundle E1 and 0![0] is nothing but

ctop(R1π∗f∗TX ) a [M].

2.4. Orbifold cohomology ring. We return to the case where X = [M/G] is a globalquotient by a cyclic group of prime order. Recall from Remark 2.2.1 that the stack IXis the disjoint union of X and copies of the G fixed locus MG indexed by nonidentityelements g ∈ G∗. The (rational) cohomology H∗(IX ) splits as a direct sum

H∗(M)G ⊕ (⊕g∈G∗H∗(MG)

).

We refer to the first summand as the untwisted sector, denoted H∗(X )Id, and the othersummands, denoted H∗(X )g, as the twisted sectors. Given cohomology classes αi ∈H∗(X )gi ⊆ H∗(IX ), i = 1, . . . , n, we define the Gromov-Witten invariant

〈α〉g,dX :=

∫

[Mg;r,u(X ,d)]vir

e∗1α1 · · · e∗nαn ∈ Q,

10 WILLIAM D. GILLAM

where the pullbacks e∗i : H∗(X )gi → H∗(Mg;r,u(X , d)) are via the evaluation mapsdefined in the previous section. We will only be interested in the case g = 0, so we willtypically drop the g from the notation.

The 3-point degree zero invariants 〈α1, α2, α3〉0X can be used as structure constants foran associative ring structure on H∗(IX ). That is, there is a unique associative, (super)-commuatative ring structure on H∗(IX ) so that

∫

Xα1α2α3 = 〈α1, α2, α3〉0X .

This ring, denoted H∗orb(X ) is called the Chen-Ruan orbifold cohomology ring.

The orbifold cohomology ring H∗orb(X ) is graded, once we shift the grading on (con-

nected components of) the twisted sectors H∗(X )g up by a (locally constant) functionage(g) defined as follows. Fix a point p ∈ Mg and consider the action of g on TpM . Theeigenvalues of this action can be written uniquely as e2π

√−1ri for some rational numbersri ∈ [0, 1). Letting (TpM)ri denote the corresponding eigenspaces, we define

age(g) :=∑

i

ri dimC(TpM)ri ∈ Q.

The function age(g) is locally constant on Mg. We say that X = [M/G] satisfies theHard Lefschetz condition if age(g) = age(g−1) for every g ∈ G. Note that this conditionis trivially satisfied if G = Z2.

Example 2.4.1. If X = [M/Z2], then the 3-point degree zero moduli spaces

M0;2,1(X , 0) = MZ2 ×BZ2

M0;0,3(X , 0) = X

are smooth of the expected dimension so their virtual fundamental classes are the usualfundamental classes; the evaluation maps are the obvious ones. The age is defined so thatthe grading on H∗(MZ2) is shifted up by half the real codimension of MZ2 in M . It followsfrom basic algebraic topology calculations that the multiplication is given as follows. Theproduct of two classes α1, α2 in the untwisted sector is the usual cup product α1α2. Theproduct of a class α in the untwisted sector and a class β in the twisted sector is theclass α|MZ2β in the twisted sector. The product of two classes β1, β2 in the twisted sectoris given by taking the usual cup product β1β2 in H∗(MZ2) and then pushing forward toH∗(M)Z2 via the inclusion MZ2 →M .

Example 2.4.2. If X = [M3/Z3] is the third alternating product of a smooth projectivevariety M of dimension d, then the fixed locus of the Z3 = A3 action is the small diagonal∆ ∼= M ⊂M3 and

H∗orb(X ) = H∗(M3)Z3 ⊕H∗−2d(M)1 ⊕H∗−2d(M)2

as a graded vector space. The stable map spaces and multiplication table are similarto those in the previous example, except when all three marked points are stacky. Forexample, the product of a class α1 ∈ H∗(M)1 and a class α2 ∈ H∗(M)2 is given by takingthe cup product in M and pushing forward to invariant cohomology H∗(M3)Z3 via theinclusion M ∼= ∆ → M3. The moduli space M0;3,0(X , 0) is isomorphic to M , but ithas expected dimension zero. There is a unique branched Z3 cover of P1 with monodromy(1, 1, 1) around the three branch points. (Indeed, such a branched cover must have genus 1


by Riemann-Hurwitz and there is only one genus 1 curve with an order 3 automorphism.)It is given by the Fermat elliptic curve E = Z(x3 + y3 + z3) ⊂ P2 with Z3 action

1 · [x : y : z] := [ζ3x : y : z],

for a primitive cube root of unity ζ3. The projection p : E → P1 given by [x : y : z] 7→ [y : z]is fully ramified at the three points

[0 : −1 : ζ3], [0 : −1 : ζ23 ], [0 : −1 : 1],

and is unramified elsewhere. The universal curve and map for M0;3,0(X , 0) = M aregiven in the diagram below.

M × Eπ1 //

²²

M ∼= ∆ ⊆M3

²²M × [E/Z3]

IdM ×p

²²π

yy

f // X

M × P1

π1

²²M

si=IdM ×[0:−1:ζi3]

;;

Letting π denote the composition of the vertical arrows, we can identify R1π∗f∗TX withthe higher equivariant pushforward R1πZ3∗ π∗1TM

3. By Serre duality we have

H1(E,OE) ∼= H0(E,ωE) ∼= Cso we compute

R1πZ3∗ π∗1TM

3 = (TM3 ⊗H1(E,OE))Z3

= (TM3)Z3

= TM.

The obstruction class is therefore given by cd(TM). It then follows that the product of twoclasses α1, α2 in the twisted sector H∗

orb(M)1 is the class α1α2 · cd(TM) ∈ H∗orb(M)2 and

similarly with sectors interchanged. One can check directly that this defines an associativemultiplication.

2.5. Potential function. We will typically assemble the GW invariants of X togetherinto an exponential generating function ΦX called the potential function. Fix a basis α0 =1, α1, . . . , αs for H∗

orb(X ) homogeneous with respect to the gradings and the splitting intosectors. For simplicity assume that the coneHeff

2 (X ) of effective homology classes on X isN〈β1, . . . , βl〉. If we can’t arrange equality here, then just take a cone containing Heff

2 (X )instead. Then ΦX is the formal power series over Q in variables x1, . . . , xs, q1, . . . , qldefined by

ΦX :=∑

d=d1β1+···+dlβl

∑

n∈Ns

〈α⊗n11 , . . . , α⊗ns

s 〉dXxn1

1 · · ·xnss

n1! · · ·ns!qd11 · · · qdl

l .


2.6. Crepant Resolution Conjecture. Finally we can state the Crepant ResolutionConjecture of Bryan and Graber [14], though we postpone the definitions of Gorensteinorbifold and crepant resolution until the next section, where they belong. For now, wejust note that if r : Y → X is any resolution of singularities, then we can take homologyclasses βl+1, . . . , βt such that the cone they generate spans the intersection of Heff

2 (Y ) andKer r∗ : H2(Y ) → H2(X).

Conjecture 2.6.1. Suppose r : Y → X is a crepant resolution of the coarse moduli spaceX of a Gorenstein orbifold X satisfying the Hard Lefschetz condition. Then there is agraded linear isomorphism

Ψ : H∗orb(X ) → H∗(Y )

and roots of unity ζl+1, . . . , ζt satisfying the following conditions.

(1) Ψ restricts to r∗ on the untwisted sector H∗(X) ⊂ H∗orb(X ).

(2) Viewing the coefficient of y1, . . . , ys, q1, . . . , qt in the potential function ΦY as apower series in ql+1, . . . , qt, this power series admits an analytic continuation from(ql+1, . . . , qt) = (ζs+1, . . . , ζt) = (0, . . . , 0) to (ql+1, . . . , qr) = (ζl+1, . . . , ζt).

(3) The potential function ΦX is obtained from ΦY via the change of variables Ψ andevaluation at (ql+1, . . . , qt) = (ζl+1, . . . , ζt), up to unstable terms.

The phrase “up to unstable terms” means that the terms in ΦY (Ψ(x))|q=ζ which havedegree at most 2 in x and degree 0 in q1, · · · , ql should be ignored. This is because themoduli spaces of degree zero stable maps to X with fewer than three marked points areempty for stability reasons, even though the moduli spaces of stable maps to Y of degreedl+1βl+1 + · · ·+ dtβt can be nonempty.

Yasuda [56] proved that there is an equality of Betti (even Hodge) numbers hi(Y ) =hi

orb(X ) for Y,X as in the conjecture. His proof uses motivic integration [27], whichKontsevich introduced in order to prove that any two crepant resolutions Y1, Y2 of X (asabove) have the same Hodge numbers. Unfortunately this only leads to an equality ofclasses in a (completed) Grothendieck ring and not to any natural identification of thecohomologies of different crepant resolutions.

The idea of the above CRC conjecture is to push the Betti/Hodge number equalityfurther. For example, one may suspect that there are ring isomorphisms H∗

orb(X ) ∼=H∗(Y ). This is not generally the case. However, the CRC does predict ring isomorphismsbetween various quantum deformations of these rings. The earliest versions of the CRC,due to Ruan [52], are roughly conjectures to this effect. In particular, the small quantumcohomology rings QH∗

orb(X ) and QH∗(Y ) should be isomorphic. Actually, there is aneven smaller quantum cohomology ring of Y , where one only deforms the multiplicationin H∗(Y ) using GW invariants for homology classes in

Ker (r∗ : H2(Y ) → H2(X)) .

Perroni [51] proved that when X has only transveral A1 and A2 surface singularities, theresulting “very small” quantum cohomology ring of Y is isomorphic to H∗

orb(X ). Werecover this result in Section 5 for global quotients X = M/Z2 as a special case of themore general Theorem 5.0.4.

We close this section by remarking that there are now more general versions of the CRC[21], [22] which should hold even when X does not satisfy the Hard Lefschetz condition.


These are more difficult to formulate and should coincide with the Bryan/Graber versionin the Hard Lefschetz case.


3. Resolutions

In this section we describe the geometry of involutions of smooth varieties and theircrepant resolutions, when they exist. Let G be a finite group acting on a smooth projective(or quasi-projective) variety M (over k) with generically trivial stabilizer. This action isGorenstein if, for any point m of M , the action of the stabilizer Gm on the fiber ωM,m istrivial. It turns out that this is equivalent to saying that the dualizing sheaf ωX for thestack quotient X = [M/G] is isomorphic to the pullback of the dualizing sheaf ωX for thecoarse quotient X = M/G. (Both dualizing sheaves are invertible for a Gorenstein action,hence the name. Indeed, the quotient scheme X is Cohen-Macaulay.) One should viewthe latter condition as asserting that X → X is a crepant resolution of singularities in thecategory of DM stacks. For example, if Z2 acts diagonally on An, n > 0, by multiplicationby −1, then the action is Gorenstein iff n is even.

We now assume that the action of G is Gorenstein. A resolution of singularities r :Y → X of the quotient X is crepant if ωY = r∗ωX . This is a sort of minimality conditionon the resolution. In general, the discrepancy divisor ωY − r∗ωX is of great importance inthe study of resolutions of (Gorenstein) singularities. Crepant resolutions do not generallyexist and are not generally unique when they do exist. However, if M is a surface, then Xhas a unique crepant resolution, and if M is a threefold, then X has at least one crepantresolution given by the G-Hilbert scheme. For Z2 actions, the situation simplifies. Itis known that the quotient of the Gorenstein involution of A2k discussed above admitsa crepant resolution iff k = 1, so we restrict our attention to involutions of M withcodimension two fixed locus. Here there is a simple description of the crepant resolution:

Theorem 3.0.2. Suppose Z2 acts on a smooth projective variety M with quotient q :M → X so that the fixed locus C := MZ2 has codimension two in M . Then there is acommutative diagram

BlCM

p

²²

q // BlCX

r

²²M

q // X

such that r is a crepant resolution and q is the quotient map for the induced Z2 action onBlCM .

Proof. This result is “well-known.” In an effort to make this as self-contained as possible,I probably made it harder than necessary.

Let IM ,IX be the ideal sheaves of C inM , X respectively. Set Y := BlCX, d := dimC.The natural map I ′

X = q−1IX ·OM → IM from the inverse image ideal sheaf ([39], Page163) induces a commutative diagram

BlI ′XM

p

²²

q // BlCX

r

²²M

q // X

respecting the Z2 actions ([39], II.7.15). To see that BlI ′XM = BlCM , it suffices to prove

that I ′X → IM induces an isomorphism onto I 2

M ⊆ IM ([39], Exer. II.7.11). This canbe checked on the local rings of points of C. By Luna’s etale slice theorem ([37], Page


198), there is a diagram, for each c ∈ C, of Z2-equivariant maps with etale vertical arrowsand cartesian squares

Mq //

²²

X

²²U

²²

// V

²²TcM // TcM/Z2

making U, V etale neighborhood of c, q(c), with Z2 acting linearly on TcM . Since C hascodimension two in M , the Z2-action on TcM has a d-dimensional 1-eigenspace and a2-dimensional (−1)-eigenspace, so the bottom arrow looks like Spec of the k-algebra map

k[x1, . . . , xd, u, v, w]/〈uv − w2〉 → k[x1, . . . , xd, y, z]u 7→ y2

v 7→ z2

w 7→ yz

with Z2 acting trivially on Spec[x1, . . . , xd]. More canonically, this is Spec of

(Sym∗k(mc/m

2c)∨)Z2 → Sym∗

k(mc/m2c)∨.

Here the map I ′X → IM corresponds to

〈u, v, w〉 → 〈y, z〉,which maps isomorphically onto 〈y, z〉2 = 〈y2, yz, z2〉 as desired.

To prove that r is a crepant resolution, we first check that it is a resolution, which canbe done etale locally. Indeed, the blowup of U along the inverse image (base change) ofC (which agrees with the blowup of U along the inverse image of IM by flatness of etalemaps) admits an etale map to BlCM making the obvious square cartesian, and similarlyfor the other etale maps. Checking that r is crepant is more tricky because one needs tocompare two dualizing sheaves and it is not so clear how to reduce this to a local question.One approach is to note that if r : Y → X is any resolution of a Gorenstein projectivevariety X, then we have ωY

∼= r∗ωX(∑

i aiEi) for some multiples ai of the exceptionaldivisors Ei of r [24]. In our case there will only be one such E and one can check thedesired equality simply by evaluating both sides on a fiber F of r, which is basically a localcomputation. (Indeed, it is not too hard to check that 〈c1(ωY ), F 〉 = 0, E.C = −2, andcertainly 〈r∗c1(ωX), F 〉 = 0, so we must have a = 0.) We will give a more self containedproof, via comparison of ideal sheaves.

Embed X in a smooth projective variety S (for example, S could be a projective spacePN

k ) and consider the commutative diagram below.

YÂ Ä j //

r

²²

B := BlCS

π

²²X

Â Ä i // S

Let I ,J be the ideal sheaves of X in S and Y in B, respectively, let r be the codimesionof X in S so C has codimension r+2 in S, and let E′ be the exceptional divisor of π. Notethat X is Cohen-Macaulay since this property is a property of local rings hence manifestly


etale local and X is etale locally a hypersurface in a smooth variety (local completeintersections are Cohen-Macaulay). By [39], Theorem III.7.11, I /I 2 and J /J 2 arelocally free sheaves of rank r on X and Y respectively and3

ωX = i∗ωS ⊗ ∧r(I /I 2)∨ωY = j∗ωB ⊗ ∧r(J /J 2)∨.

Note that the left map in the exact sequence

I /I 2 → i∗ΩS → ΩX → 0

is monic away from C so we have i∗ωS |X\C = ∧d+2ΩX ⊗ ∧r(I /I 2) and hence ωX\X =∧d+2ΩX\C . Since B is a blowup along the smooth codimension (r + 2) subvariety C, wehave

ωB = π∗ωS ⊗OB((r + 1)E′)

by [36], Page 608. So

r∗ωX = r∗i∗ωS ⊗ r∗ ∧r (I /I 2)∨

= j∗π∗ωS ⊗ r∗ ∧r (I /I 2)∨ωY = j∗π∗ωS ⊗ j∗OB((r + 1)E′)⊗ ∧r(J /J 2)∨

and hence it suffices to prove that

r∗ ∧r (I /I 2)⊗ j∗OB((r + 1)E′) ∼= ∧r(J /J 2).

The commutativity of the diagram implies that π−1I · OB ⊆ J so we have a naturalmap

r∗ ∧r (I /I 2) → ∧r(J /J 2)

which we can tensor with j∗OB((r + 1)E′) to get a map

j∗OB((r + 1)E′)⊗ r∗ ∧r (I /I 2) → j∗OB((r + 1)E′)⊗ ∧r(J /J 2).

I claim this is an isomorphism onto

∧r(J /J 2) ⊆ ∧r(J /J 2)⊗ j∗OB((r + 1)E′).

This can be checked etale locally, where our embedding X → S looks like Spec of a ringmap

k[x, u, v, w, z1, . . . , zr−1] ³ k[x, u, v, w]/〈uv − w2〉zi 7→ 0.

Then the blowup BlCS is the closed subset of

Spec k[x, u, v, w, z1, . . . , zr−1]× Proj k[U, V,W,Z1, . . . , Zr−1]

given by the vanishing of the 2× 2 minors of the matrix below.(U V W Z1 · · · Zr−1

u v w z1 · · · zr−1

)

3Technically, III.7.11 only proves this in case S = PNk , but the same result holds with S any smooth

projective variety as one sees easily by applying the adjunction formula to the obvious conormal sequences.


On the open set D+(U), we have local coordinates S = V/U , T = W/U , τi = zi/u andour commutative square looks like Spec of the diagram below.

uτi uS uT

zi_

OO

v_

OO

w_

OO k[x, u, S, T, τ1, . . . , τr−1]

²²

// k[x, u, S, T ]/〈S − T 2〉

²²k[x, u, v, w, z] // k[x, u, v, w]/〈uv − w2〉

In particular, r is a resolution in this patch. In these coordinates the exceptional locusE′ is given by V (u), so j∗OB((r + 1)E′) is free with generator u−(r+1) and I /I 2 andJ /J 2 are free on (uv − w2), z1, . . . , zr and (S − T 2), τ1, . . . , τr−1 respectively. Our mapis given by

u−(r+1) ⊗ (uv − w2) ∧ z1 ∧ · · · ∧ zr−1 7→ u−(r+1) · (u2S − u2T 2) ∧ uτ1 ∧ · · · ∧ uτr−1

= (S − T 2) ∧ τ1 ∧ · · · ∧ τr−1

hence it is as claimed. The computations in the other coordinate patches are similar. ¤

We now state some other facts about the geometry of our resolutions. We will not provethese, but we note that they can all be established etale locally as above.

There is a SES0 → Sym2

OCIM/I

2M → ΩX |C → ΩC → 0

of locally free sheaves on C. In particular, ΩX |C is locally free of rank d+ 3 on C despitethe fact that ΩX is locally free of rank d+ 2 on X \ C.

The exceptional divisor E of r is a conic (bundle) in the P2-bundle P Sym2NC/M overC. Indeed, the closed embedding E → PSym2NC/M is obtained by projectivizing the(dual of the) diagonal morphism of locally free sheaves IM/I

2M → Sym2 IM/I

2M .

The quotient X is manifestly an etale V -variety. Setting ΩpX := qZ2∗ (∧pΩM ) we have

ωX = Ω2X so that, in particular, Ω2

X is invertible. However, Ω1X is not locally free.

The normal bundle to E in BlCM is OE(−1) (c.f. [39], II.8.2.4(c), [36], Page 607) andhence the normal bundle to E in Y is OE(−2). Indeed, if Z2 acts on a smooth variety Vwith fixed locus E of codimension 1, then the quotient is smooth and the normal bundleof E in the quotient is naturally isomorphic to the second symmetric power of the normalbundle of E in V .

One can also prove that the crepant resolution r : Y → X described above is the uniquecrepant resolution: this more or less follows from the universal property of blowing up.


4. Local Theory

In this section we use localization techniques in the style of R. Cavalieri (who followsR. Pandharipande and C. Faber) to compute some new Hodge integrals on the modulispace of genus zero twisted stable maps to BZ2 as defined in Section 2. This is a compacti-fication of the moduli space of elliptic curves (with marked branch points) so these integralscan be viewed as Hurwitz-Hodge integrals over the (closure of the) locus of hyperellipticcurves Hg ⊆ Mg. We include a self-contained computation of the Hodge integrals of theform λgλg−iψ

i−1, which is not at all original, but which we have assembled from variouspapers of Cavalieri for the reader’s convenience and for later use. In the last part of thissection, we generalize this to compute all Hodge integrals of the form λgλg−iψ

k11 · · ·ψk2g+2

2g+2 .

Recall the definition of M0;2g+2,n(BZ2) from Section 2. An object of this stack over ascheme B is a diagram

C

p

²²C

π

²²B

Ri

99

Ui

II

where C is a double cover of C branched along the Ri and possibly at nodes of C, butnot elsewhere. Various flat, proper, stability, and balancing conditions are imposed as inSection 2. The curve C is required to have arithmetic genus zero, hence C is manifestlyhyperelliptic of arithmetic genus g by the Riemann-Hurwitz formula.

The stack M0;2g+2,n(BZ2) is the same as the stack of admissible covers (or rather, itsnormalization) called

Admg→0; (2)1,...,(2)2g+2,(1+1)1,...,(1+1)n

by Cavalieri in [18] and [19]. See [3] for discussion.

There are forgetful maps

M0;2g+2,n(BZ2)F−−−−→ Mg;2g+2

G

yM0;2g+2+n

with F given by forgetting C and G given by forgetting C.

We use F and G to define vector bundles and cohomology classes on M0;2g+2,n(BZ2).On Mg;2g+2 we have the Hodge bundle E (pulled back from Mg) whose fiber over Cis H0(C, ωC); we define λi ∈ H2i(M0;2g+2,n(BZ2)) to be ci(F ∗E). On M0;2g+2+n wehave the cotangent line bundles Li with fiber T ∗Pi

C over (C,P1, . . . , P2g+2+n); we defineψi := c1(G∗Li). It is natural to pull back the cotangent line classes by G, rather than byF , because we will be interested in the normal bundle in M0;2g+2,n(BZ2) of various lociof nodal curves. That is, we will be interested in infinitesimal deformations of brancheddouble covers (C, C) (as opposed to deformations of C as a genus g curve), which aredetermined by deformations of the base curve C because G is an etale map of stacks of


degree 1/2. Of course, for p ∈ Ri ⊂ C, the cotangent space in the Z2 quotient T ∗pC isequal to T ∗p (C)⊗ T ∗p (C) so we have G∗ψi = 2F ∗ψi for i = 1, . . . , 2g + 2.

The composition of F and the forgetful map Mg;2g+2 → Mg gives a finite map ofdegree (2g + 2)! from M0;2g+2,0(BZ2) onto the closure Hg of the hyperelliptic locus inMg. (When g = 1, one of the marked points should be retained.) It is probably worthdeciding whether the natural maps

[M0;4,0/S3] →M1,1

[M0;6,0(BZ2)/S6] →M2

are equivalences of stacks, but we will not pursue this here.

We make the identification H∗C∗(pt) = C[t] with t equal to the equivariant first Chern

class of a weight 1 action of C∗ on C in the equivariant cohomology of a point. For aC∗-equivariant vector bundle F on a C∗-space, let F [n] denote the C∗-equivariant vectorbundle obtained by tensoring F with a trivial bundle acted on with weight n. We willfrequently use the standard notation

Λg(n) := cg(E[n]) =g∑

i=0

(nt)iλg−i = (−1)gcg(E∨[−n]).

We will also frequently make use of Mumford’s Relation [46]: c(E⊕ E∨) = 1.

In this section we will compute the following integrals:

Dg,i :=∫M0;2g+2,0(BZ2) λgλg−iΨi−1 (D0,0 := 1/2)

Lg :=∫M0;2g+2,0(BZ2)

∑gi=1 λgλg−iΨi−1 (L0 := 1/2)

=∑g

i=0Dg,i

M+g :=

∫M0;2g+2,0(BZ2)

Λg(1)Λg(2)t−Ψ (M+

0 := 1/2)

=∫M0;2g+2,0(BZ2)

Λg(−1)Λg(−2)−t−Ψ

M−g :=

∫M0;2g+2,0(BZ2)

Λg(1)Λg(2)−t−Ψ (M−

0 := 1/2)

=∫M0;2g+2,0(BZ2)

Λg(−1)Λg(−2)t−Ψ

N+g :=

∫M0;2g+2,1(BZ2)

tΛg(1)Λg(2)t−Ψ2g+3

(N+−1 := 1/4)

=∫M0;2g+2,1(BZ2)

tΛg(−1)Λg(−2)t+Ψ2g+3

N−g :=

∫M0;2g+2,1(BZ2)

tΛg(1)Λg(2)t+Ψ2g+3

(N−−1 := 1/4)

=∫M0;2g+2,1(BZ2)

tΛg(−1)Λg(−2)t−Ψ2g+3

Notice that each integrand is of the correct degree to integrate. It is understood that theC∗ action is trivial so the t variables are just placeholders. One can expand the integrandusing the usual 1

1−x power series, so that, for example,

M+g =

∫(λg + tλg−1 + · · ·+ tg)(λg + 2tλg−1 + · · ·+ 2gtg)(t−1 + t−2Ψ + . . . )

=∫ 2g∑

i=1

(i∑

j=0

2jλg−jλg+j−i)Ψi−1.

The integrals Dg,1 were first computed by Faber-Pandharipande [31]. The Dg,i (and henceLg) were first computed by Cavalieri (in considerably more generality). The Lg integrals


were computed in [18], and a recursion (our 1 below) expressing the Dg,i in terms ofthe Dg,1 was determined in [19]. It was noted in [10] that these could be combined todetermine all the Dg,i, as we have done below.

The rest of these appear to be new. It is not clear how to compute the individual termsin, say, the M+

g sum, but in practice it turns out that the entire sum is often what arises,as we shall soon see.

4.1. Generating Functions and Recursions. We assemble the above rational numbersinto the following generating functions:

Di :=∑

g≥i

Dg,ix2g

(2g)!

L :=∞∑

g=0

Lgx2g+1

(2g + 1)!

F :=∞∑

i=0

(−2)iDi

M+ :=∞∑

g=0

M+g

x2g+1

(2g + 1)!

M− :=∞∑

g=0

M−g

x2g+1

(2g + 1)!

N+ :=∞∑

g=−1

N+g

x2g+2

(2g + 2)!

N− :=∞∑

g=−1

N−g

x2g+2

(2g + 2)!

Notice that4

L =∞∑

i=0

Di

andN+

0 = N−0 =

∫

BZ2

1 = 1/2.

We will prove the following relations in the next section:

For i > 1 : Dg,i = −2g∑

g1=0

(2g − 12g1 − 1

) i∑

k=1

(−1)kDg1,i−kDg−g1,k(1)

Lg =12g

g−1∑

i=0

(−1)g−i+1

(2g + 1

2i

)Li(2)

Lg = 2g∑

g2=0

(2g + 12g2

) g2∑

i=0

(−2)iDg2,iM+g−g2

(3)

4Dots denote derivatives with respect to x.


g∑

g1=1

(2g

2g1 − 1

)M+

g1M−

g−g1= 2

g∑

g1=0

(2g2g1

)N+

g1N−

g−g1−1(4)

g∑

g1=0

(2g2g1

)M+

g1M−

g−g1= 2

g−1∑

g1=0

(2g

2g1 + 1

)N+

g1N−

g−g1−1(5)

g∑

g2=1

(2g

2g2 − 1

)M+

g−g2M−

g2= 2

g∑

g2=0

(2g2g2

)N+

g−g2−1N−g2

(6)

We now argue that these relations uniquely determine all the numbers under consid-eration. Surely 2 determines all the Lg. Then 1 expresses the Dg,i in terms of Dg,1 andDg1,k with g1 < g, so, since Lg is the sum of the Dg,i this determines Dg,1 and hence therest of the Dg,i. Then 3 determines the M+

g recursively since M+g appears on the RHS

with coefficient 1 when g2 = 0. Next 4 determines N+g in terms of M+

g and symbols withsmaller subscript, 5 determines M−

g in terms of M+g and symbols with smaller subscript,

and finally 6 determines N−g in terms of M−

g and symbols with smaller subscript.

Expressing these in terms of generating functions we get:i∑

k=0

(−1)kDi−kDk = 0(7)

L sinx = L(8)2M+F = L(9)

M+M− + 1/4 = 2N

+N−(10)

M+M

−= 2N

+N−

+ 1/4(11)

M−M+ + 1/4 = 2N

−N+(12)

One checks easily that 7 implies

Di =2i−1

i!(D1)i(13)

hence we find that

L =∞∑

i=0

Di =12

∞∑

i=0

2i

i!(D1)i =

12

exp(2D1)

and

F =∞∑

i=0

(−2)iDi =∞∑

i=0

(−2)i 2i−1

i!(D1)i =

12

exp(−4D1).

Solving 8, then plugging in to the above equations and using 9 we get:

L = tan(x/2)(14)D1 = ln sec(x/2)(15)F = (1/2) cos4(x/2)(16)

M+ = tan(x/2) sec4(x/2)(17)

=x

2+

74x3

3!+

474x5

5!+

2512x7

7!+ . . .(18)


At this point we are faced with the system of ODEs 10-12, 17. If we massage things alittle bit, we can decouple this system and solve the resulting ODEs. Summing 10, twice11, and 12 and subtracting 1/2 from both sides we get

M+M− + 1/8 = 2N+N−(19)

(we actually get the second derivative of 19, but 19 follows by taking initial conditionsinto account). Considering 10 minus 12 we get

M−M+ −M+M

−= 2N−N

+ − 2N+N−

(20)

then we differentiate 19 and add and subtract the result to 20 to get:

M−M+

= 2N−N+

(21)

2N+N−

= M+M−

(22)

Now multiply 22 by M+

and use 11 to get

2N+M+N−

= M+(2N+N−

+ 1/4)

and divide by M+N+N−

and use 22 to get:

2M

+

M+ = 2N

+

N+ +1

2M+M−(23)

Now divide 21 by 2N+N− and use 19 to get

N+

N+ =M−M

+

M+M− + 1/8(24)

and then plug 24 into 23 and simplify to get:

M+M

−= 2M+M− + 1/4(25)

Finally, we plug 25 back into 24 and use the symmetry in exchanging +/− superscriptseverywhere to get

N+M

−= 2M−N+(26)

N−M

+= 2N−M+(27)

N+N−

= 2N+N− − 1/8(28)

(28 follows from 25 using 11 and 19). Plugging in for M+, 27 reads(2 tan2 x

2sec4 x

2+

12

sec6 x

2

)N−

= 2N− tanx

2sec4 x

2.(29)

We can solve this explicitly:

N− = (1/4)(3− 2 cosx)(30)

so that, for g ≥ 0, N−g = (−1)g/2. Plugging in for M+ in 25 we get:

(2 tan2 x

2sec4 x

2+

12

sec6 x

2

)M

−= 2M− tan

x

2sec4 x

2+

14

(31)

The solution to this is given by

M− = (1/32)(9x− 6x cosx+ 14 sinx− cosx sinx)(32)

=x

2+

14x3

3!− x5

5!+

238x7

7!+ . . .(33)


Finally, we can plug in our solution for N− into 28 and multiply by 8 to get

4N+

sinx = 4N+(3− 2 cosx)− 1(34)

which has solution:

N+ = csc2 x tan2 x

2+

14

cscx tan3 x

2+

34x csc2 x tan3 x

2

=14

+12x2

2!+

52x4

4!+

432x6

6!+

5572x8

8!+ . . .

Example 4.1.1. On M0;4,0(BZ2), Mumford’s relation implies 2λ2 = λ21 and, upon ex-

panding D1 and D2, we find λ2λ1 = 18 and λ2ψ = 3

8 (integration symbols suppressed).We also have

74

= M+2 = 3λ2λ1 + 5λ2ψ + 2λ2

1ψ + 6λ1ψ2 + 4ψ3.

All of the terms in this sum are determined by Mumford’s relation and the two valuesabove.

4.2. Proofs of the Recursions via Localization. All of the recursions described inthe previous section are arrived at by computing integrals on the “graph space”

M0;2g+2,0(P1 ×BZ2, 1)

using localization. This moduli space parameterizes genus 0 stable maps g : Σ → P1 ofdegree 1 (essentially a choice of distinguised component of Σ and an isomorphism fromthis component to P1), together with an admissible degree 2 cover Σ of the domain curveΣ. There are natural evaluation maps

ei : M0;2g+2,0(P1 ×BZ2, 1) → P1

(g : Σ → P1, Σ, R1, . . . , R2g+2) 7→ g(Ri).

Let U, U be the universal genus 0 stable map of degree 1, and its universal admissibleZ2-cover, respectively. Let f, f be the corresponding universal maps, and let π be theprojection from U to M0;2g+2,0(P1 ×BZ2, 1).

Let C∗ act on a 2-dimensional vector space V with weights 0 and 1 with correspond-ing weight-space basis v0, v1. Then C∗ acts naturally on P1 = PV (and hence also onM0;2g+2,0(P1 × BZ2, 1)) with fixed points 0 := 〈v0〉 ∈ P1 and ∞ := 〈v1〉 ∈ P1, and T0P1

and T∞P1 are acted on with weights 1,−1, respectively. The natural lifting of OP1(−1)has weights 0, 1, but we can tensor with OP1 [n] to get a different linearization with weightsn, n+ 1 over 0,∞, respectively.

The fixed locusM0;2g+2,0(P1×BZ2, 1)C∗

is a disjoint union of smooth DM stacks indexedby the set A := i ∈ 1, . . . , 2g + 2 : ei = 0. Given any class α ∈ H∗

C∗(M0;2g+2,0(P1 ×BZ2, 1)), the equivariant push-forward

∫

M0;2g+2,0(P1×BZ2,1)α

to H∗C∗(BZ2) = C[t] can be computed as a sum of integrals over the fixed loci by restricting

α and dividing by the Euler class of the normal bundle to each fixed component. Thefixed loci and their contributions to this equivariant integral are described in the tablesbelow.


To compute the weights in the normal direction of each fixed locus, one must considerthe Z2-invariant deformations of

(g : Σ → P1, Σ, R1, . . . , R2g+2)

Type-I Fixed Loci|A| = 2g + 1 (codim 3) |A| = 1 (codim 3)

0 P1

Σ1

Σ1

∞ 0 P1 ∞

Σ2

Σ2

∫M0;2g+2,0(BZ2)

i∗(α)(t)(−t)(t−Ψ)

∫M0;2g+2,0(BZ2)

i∗(α)(t)(−t)(−t−Ψ)

|A| = 2g1 + 1 (codim 4) (1 ≤ g1 ≤ g − 1)

0 ∞P1

Σ2Σ1

Σ1 Σ2

2∫M0;2g1+2,0(BZ2)×M0;2g−2g1+2,0(BZ2)

i∗(α)(t)(−t)(t−Ψ1)(−t−Ψ2)

Type-II Fixed Loci|A| = 2g + 2 (codim 2) |A| = 0 (codim 2)

P10

Σ1

Σ1

∞ 0 ∞P1

Σ2

Σ2

∫M0;2g+2,1(BZ2)

i∗(α)(t)(−t)(t−Ψ)

∫M0;2g+2,1(BZ2)

i∗(α)(t)(−t)(−t−Ψ)

|A| = 2g1 + 2 (codim 4) (0 ≤ g1 ≤ g − 1)

P10 ∞

Σ1

Σ2

Σ2

Σ1

2∫M0;2g1+2,1(BZ2)×M0;2g−2g1,1(BZ2)

i∗(α)(t)(−t)(t−Ψ1)(−t−Ψ2)


which leave the fixed locus; these are just deformations of (g,Σ) leaving the fixed locus.For example, when |A| = 2g1 + 1 with 1 ≤ g1 ≤ g − 1, the deformations leaving the fixedlocus are moving the collapsed component over 0 and ∞ (these correspond infinitesimallyto T0P1 and T∞P1 respectively, hence the (t) and (−t)) and smoothing the nodes N0 andN∞ over 0 and ∞ (these correspond to TN0Σ1⊗T0P1 and TN0Σ2⊗T∞P1). We have abusednotation a little bit in these tables: the Ψ classes are supposed to be the cotangent lineclasses at the marked point where the collapsed curve is to be glued to the non-collapsedcurve. In particular, the Ψ classes appearing in the Type-II contribution classes are thecotangent line classes at the marked non-ramification points. The factor of 2 in front ofthe bottom contributions has to do with counting the size of automorphism groups andthe number of ways to glue.

Proof of 1. We compute∫

M0;2g+2,0(P1×BZ2,1)c1(e∗1O(1))c1(e∗2O(1))c1(e∗2g+2O(1))cg(R1π∗f∗O)cg−i(R1π∗f∗O),

which vanishes for dimension resons when i > 1, using the linearizations below.

Bundle Weight over 0 Weight over∞O(1) 1 0O(1) 1 0O(1) 0 −1O 0 0O 0 0

Because of the choice of weights on the O(1)’s, only fixed loci with 1, 2 ∈ A and 2g+2 /∈ Awill contribute to the localization sum; in particular, only the Type-II loci where thereare collapsed components over 0 and ∞ might contribute. However, these Type-II lociactually do not contribute because the restriction of the Hodge bundle (or rather its dual)to such a locus will contain a trivial summand (which will even be equivariantly trivialsince the weights on the two O’s are 0), as we can see by taking the long exact cohomologysequence associated to the normalization sequence:

0 → OΣ → O2P1 ⊕OΣ1

⊕OΣ2→ C4 → 0.

Similarly, from examining the normalization sequence, we find that on the fixed locuswhere A = 1, . . . , 2g + 1, we can identify R1π∗f∗O with E∨. So the contribution to thelocalization sum from this locus is∫

M0;2g+2,0(BZ2)

(t)(t)(−t)(−1)iλgλg−i

(t)(t−Ψ)(−t) =(−1)iDg,i

ti−1.

Now we consider the contribution from a fixed locus where |A| = 2g1+1 (g1 = 1, . . . , g−1),with 1, 2 ∈ A, 2g + 2 /∈ A. Examining the normalization sequence, we see that H1(Σ,OΣ)splits as H1(Σ1,OΣ1

)⊕H1(Σ2,OΣ2) so we have

cg(R1π∗f∗O)cg−i(R1π∗f∗O) = (−1)icg(Eg1 ⊕ Eg−g1)cg−i(Eg1 ⊕ Eg−g1)

= (−1)i(λg1λg−g1)(λg1−iλg−g1 + · · ·+ λg1λg−g1−i)

and hence the contribution from this locus is

2∫

(t)(t)(−t)(−1)i(λg1λg−g1)(λg1−iλg−g1 + · · ·+ λg1λg−g1−i)(t)(t−Ψ1)(−t)(−t−Ψ2)

,


where the integral is over the product M0;2g1+2,0×M0;2g−2g1+2,0. Expanding the denomi-nators as power series and collecting only the terms of the correct degree to integrate overthe product, we can write the sum of the contributions from all such loci as

2(−1)i

ti−1

g−1∑

g1=1

(2g − 12g1 − 1

) i∑

k=1

(−1)kDg1,i−kDg−g1,k

(the binomial coefficient just counts the number of such A). We might as well make this asum over g1 = 0, . . . , g, because when g1 = 0 the binomial coefficient vanishes, and wheng1 = g, all the terms Dg−g1,k vanish. Since the sum of all the contributions is 0 when i > 1,we obtain 1 by multiplying through by (−1)iti−1, and solving for the first contribution.

Proof of 2. We compute the “Calabi-Yau cap integral”∫

M0;2g+2,0(P1×BZ2,1)c1(e∗1O(1))c2g+1(R1π∗f∗(O ⊕O(−1))

using the two linearizations below. Since the integrand is of the correct degree, the valueof the integral is independent of the chosen linearization.

Linearization A Linearization BBundle Weight over 0 Weight over∞ Weight over 0 Weight over∞O(1) 0 −1 0 −1O 0 0 1 1

O(−1) −1 0 −1 0

First of all, in both cases, the choice of linearization of O(1) implies that only loci with1 /∈ A will contribute. Examining the normalization sequence and noting that O(−1) hasweight 0 over ∞ in both linearizations we see that the Type-II fixed loci do not contribute.For a Type-I fixed locus with |A| = 2g1 +1 < 2g+1 in Linearization A, the normalizationsequence shows

H1(Σ,OΣ ⊗ g∗OP1(−1)) = H1(P1,O(−2))⊕H1(Σ1,OΣ1)[−1]⊕H1(Σ2,OΣ1

)

H1(Σ,OΣ) = H1(Σ1,OΣ1)⊕H1(Σ2,OΣ1

).

The bundle with fiber H1(P1,O(−2)) is of course trivial on this locus, but not equiv-ariantly. Its weight can be worked out using an explicit Cech cocycle representative forH1(P1,O(−2)), to find c1(H1(P1,O(−2))) = −t/2, so on this locus we have

c2g+1(R1π∗f∗(O ⊕ O(−1))) = (−t/2)Λg1(1)λg1λg−g1λg−g1 ,

but λg−g1λg−g1 = 0 by Mumford’s relation, so these loci will not contribute to the local-ization sum. Thus the only contributing locus is the one with A = 2, 3, . . . , 2g+2, whichcontributes ∫

M0;2g+2,0(BZ2)

(−t)(−t/2)λgΛg(1)(t)(−t)(t−Ψ)

=−12Lg.

Again making use of the normalization sequence, we can work out the contributions inLinearization B.

The Linearization B contribution from A = 2, 3, . . . , 2g + 2 is∫

M0;2g+2,0(BZ2)

(−t)(−t/2)Λg(1)Λg(−1)(t)(−t)(t−Ψ)

.


Now, notice that

Λg(1)Λg(−1) = (λg + tλg−1 + · · ·+ tg)(λg − tλg−1 + · · ·+ (−1)gtg)

=2g∑

i=0

(i∑

k=0

(−1)kλg−kλg+k−i

)ti

= (−1)gt2g,

because the term inside the parenthesis vanishes by Mumford’s relation unless i = 2g, k =g. Also recall that ∫

M0;2g+2

Ψ2g−1 = 1,

so, since our Ψ is the pullback of the Ψ class on M0;2g+2 by the etale degree 1/2 map G,we have ∫

M0;2g+2,0(BZ2)Ψ2g−1 =

12,

so the above contribution is just

−12

∫

M0;2g+2,0(BZ2)

(−1)gt2g

t−Ψ=−12

∫

M0;2g+2,0(BZ2)(−1)gΨ2g−1 =

−12

(−1)gL0

(recall that we defined L0 := 1/2).

Similarly, for 1 ≤ g1 ≤ g− 1, the contribution from all the fixed loci with |A| = 2g1 + 1will be

2(

2g + 12g1 + 1

) ∫(−t)(−t/2)Λg1(1)Λg1(−1)λg−g1Λg−g1(−1)

(t)(−t)(t−Ψ1)(−t−Ψ2)

=(

2g + 12g1 + 1

) ∫Λg1(1)Λg1(−1)λg−g1Λg−g1(−1)

(t−Ψ1)(t+ Ψ2)

=(

2g + 12g1 + 1

)(∫(−1)g1Ψ2g1−1

)(∫λg−g1(λg−g1 − tλg−g1−1 + · · ·+ (−1)g−g1tg−g1)

t+ Ψ2

)

=−12

(−1)g1

(2g + 12g1 + 1

)Lg−g1

and the contribution from all the loci with |A| = 1 will be

(2g + 1)∫

M0;2g+2(BZ2)

(−t)(−t/2)λgΛg(−1)(−t)(t)(−t−Ψ)

=−12

(2g + 1)Lg.

Thanks to our convention L0 = 1/2 we can write the entire Linearization B computationof the Calabi-Yau cap integral as

−12

(2g + 1)Lg − 12

g∑

g1=1

(−1)g1

(2g + 12g1 + 1

)Lg−g1 .

Equating the two localization computations, multiplying by −2, solving for Lg, and rein-dexing the sum with i = g − g1 yields 2.


Proof of 3. Compute the Calabi-Yau cap integral using the linearizations below.

Bundle Weight over 0 Weight over∞O(1) 0 −1O −2 −2

O(−1) −1 0

Since O(−1) is linearized with 0 weight over ∞ and the weight of O(1) is 0 over 0, itfollows from examining the normalization sequence that only Type-I loci with 1 /∈ A willcontribute to the localization sum. When A = 2, . . . , 2g + 2, the contribution is

∫

M0;2g+2,0(BZ2)

Λg(1)Λg(2)(−t)(−t/2)(t)(−t)(t−Ψ)

= −12M+

g

and when 2g + 2− |A| = 2g2 + 1 > 1, the contribution is

2(

2g + 12g2

) ∫Λg−g2(1)Λg−g2(2)(−t)(−t/2)λg2Λg2(2)

(t)(−t)(t−Ψ)(−t−Ψ)

=(

2g + 12g2

) ∫Λg−g2(1)Λg−g2(2)

(t−Ψ)

∫λg2Λg2(2)(t+ Ψ)

= −(

2g + 12g2

)M+

g−g2

(g2∑

i=0

(−2)iDg2,i

)

The contribution from all the fixed loci where |A| = 1 is given by

(2g + 1)∫

M0;2g+2,0(BZ2)

λgΛg(2)(−t)(−t/2)(−t)(t)(−t−Ψ)

= −(2g + 1)2

g∑

i=0

(−2)iDg,i.

We also know from the above localization computation that the value of the Calabi-Yaucap integral is −(1/2)Lg. Equating the two localization computations, multiplying by −2,and taking the initial values D0,0 = 1/2 and M+

0 = 1/2 into account yields 3.

Proof of 4. We compute the integral∫

M0;2g+2,0(P1×BZ2,1)c1(e∗1O(1))c1(e∗2O(1))c2g(R1π∗f∗O ⊕O)

using the linearization below.

Bundle Weight over 0 Weight over∞O(1) 1 0O(1) 1 0O −1 −1O −2 −2

First of all, notice that this integral is 0 by Mumford’s relation (take weight 0 on bothO’s). However, with this linearization we will see a contribution from Type-I and Type-IIfixed loci. The weights on the O(1)’s force any contributing locus to have 1, 2 ∈ A. Thecontribution from the locus where A = 1, . . . , 2g + 2 is

∫

M0;2g+2,1(BZ2)

(t)(t)Λg(1)Λg(2)(t)(t−Ψ)

= N+g .


For a locus where |A| = 2g1 + 2 < 2g + 2, we observe that the genera of the curvecollapsed over 0 and the curve collapsed over ∞ do not add to g, and the normalizationsequence shows that the dual of the Hodge bundle contains a trivial (but not equivariantly!)summand, whose weight is just the weight we assigned to O above. Thus the contributionfrom all such loci will be

2(

2g2g1

) ∫(t)(t)Λg1(1)Λg1(2)Λg−g1−1(1)Λg−g1−1(2)(−t)(−2t)

(t)(−t)(t−Ψ1)(−t−Ψ2)

= 4(

2g2g1

)N+

g1N−

g−g1−1.

This explains our convention N−−1 := 1/4: the formula above is correct even when g1 = g.

For the Type-I loci, the contribution from the loci with |A| = 2g + 1 will be

(2g)∫

M0;2g+2(BZ2)

(t)(t)Λg(1)Λg(2)(t)(−t)(t−Ψ)

= −2gM+g

and the contribution from the loci with |A| = 2g1 + 1 < 2g + 1 will be

2(

2g2g1 − 1

) ∫(t)(t)Λg1(1)Λg1(2)Λg−g1(1)Λg−g1(2)

(t)(−t)(t−Ψ1)(−t−Ψ2)

= −2(

2g2g1 − 1

)M+

g1M−

g−g1.

Again, our convention M−0 = 1/2 is convenient since the above formula is correct even

when g1 = g. The total integral is 0, so dividing everything by 2 and solving for the Type-Icontributions yields 4.

Proof of 5. Compute the same integral as above, but with weights as below.

Bundle Weight over 0 Weight over∞O(1) 1 0O(1) 0 −1O −1 −1O −2 −2

These weights are chosen so that only Type-II loci where there are collapsed curves over0 and ∞ will contribute. The contribution from the locus where A = 1, 3, 4, . . . , 2g+ 2is ∫

M0;2g+2,0(BZ2)

(t)(−t)Λg(1)Λg(2)(t)(−t)(t−Ψ)

= M+g

and the contribution from the locus where A = 1 is∫

M0;2g+2,0(BZ2)

(t)(−t)Λg(1)Λg(2)(t)(−t)(−t−Ψ)

= M−g .

The contribution from all the loci where |A| = 2g1 + 1 (g1 = 1, . . . , g − 1) is

2(

2g2g1

) ∫(t)(−t)Λg1(1)Λg1(2)Λg−g1(1)Λg−g1(2)

(t)(−t)(t−Ψ1)(−t−Ψ2)

= 2(

2g2g1

)M+

g1M−

g−g1


and the contribution from the loci where |A| = 2g1 + 2 (g1 = 0, . . . , g − 1) is

2(

2g2g1 + 1

) ∫(−t)(t)(−t)(−2t)Λg1(1)Λg1(2)Λg−g1−1(1)Λg−g1−1(2)

(t)(−t)(t−Ψ1)(−t−Ψ2)

= −4(

2g2g1 + 1

)N+

g1N−

g−g1−1.

Since the total integral is 0, we may divide by 2 and solve for the Type-I contributions toget 5.

Proof of 6. Compute the same integral one last time using the weights below.

Bundle Weight over 0 Weight over∞O(1) 0 −1O(1) 0 −1O −1 −1O −2 −2

The contributions of the fixed point loci can be worked out as above; we omit the details.

4.3. More ψ classes. In this section we extend the previous calculations to determineHurwitz-Hodge integrals of the form λgλg−iψ

k11 · · ·ψk2g+2

2g+2 . Let m = (m1, . . . ,ml) be anordered l-tuple of nonnegative integers. Write [l] for 1, . . . , l, |m| for m1 + · · ·+ml, andψm for ψm1

1 · · ·ψmll . Let

Dg;m,i =∫

M0;2g+2,0(BZ2)λgλg−iψ

mψi−|m|−1 ∈ Q

so that our previously defined Dg,i is now Dg;0,i, where 0 is an tuple with zeroes as entries,or the empty tuple. Assemble these into generating functions

Dm,i :=∑

g≥i

Dg;m,ix2g

(2g)!.

Consider the integral∫

M0;2g+2,0(P1×BZ2,1)ψm1

1 · · ·ψmll (e∗2gh) (e∗2g+1h) (e∗2g+2h) cg(R

1π∗f∗O)cg−i(R1π∗f∗O),

where the hyperplane classes h = c1(OP1(1)) and R1π∗f∗O bundles are linearized as inthe proof of Relation 1, so the first two vanish at ∞ and the third vanishes at 0.

This integral vanishes for dimension reasons as long as 3+ |m|+2g− i < 2g+2, that is,as long as |m|+1 < i. No fixed locus of Type I contributes to the localization sum for thisintegral because of the usual argument: the linearization of the h’s forces the componentscollapsed over 0 and ∞ to be connected, and hence cg(R1π∗f∗O) will vanish on a Type Icomponent since its graph contains a loop.

When A = 1, . . . , 2g + 1, the fixed locus is just M0;2g+2,0(BZ2) and its contributionis (the integral of)

(−1)i(t)(t)(−t)ψmλgλg−i

t(−t)(t− ψ)=

1ti−|m|−1

(−1)iλgλg−iψmψi−|m|−1.

(Here the ψ class with no subscript is the one at the gluing marked point.)


To write the contribution from a general fixed Type II fixed locus, let B denote thesubset of [l] consisting of those i such that ei 7→ ∞ and let B := [l] B. If B = b1, . . . , bsthen write m(B) for the s-tuple (mb1 , . . . ,mbs). The contribution from such a fixed locuswith genus g1 over 0 is then

2(−1)i(t)(t)(−t)ψmλg1λg−g1(λg1λg−g1−i + · · ·+ λg1−iλg−g1)

(t)(−t)(−t− ψ)(t− ψ)

= 2(−1)i

ti−|m|−1

i∑

k=0

(−1)k−|m(B)|(λg1λg1−(i−k)ψ

m(B)ψi−k−|m(B)|−1)

·(λg−g1λg−g1−kψ

m(B)ψk−|m(B)|−1)

= 2(−1)i

ti−|m|−1

i∑

k=0

(−1)k−|m(B)|Dg1;m(B),i−kDg−g1;m(B),k

where the integration over M0;2g1+2,0(BZ2) and M0;2(g−g1)+2,0(BZ2) is suppressed in thefirst and second parentheses, respectively. The total number of fixed loci with genus g1over zero for a fixed B is given by

(2g − 1− l

2g1 − 1− |B|).

Since the integral we are computing vanishes (when |m| < i − 1) we can get rid of theleading factor 2(−1)it|m|+1−i to arrive at the recursion

∑

B⊆[l]

∑

k

(−1)k+|m(B)|∑g1

(2g − 1− l

2g1 − 1− |B|)Dg1;m(B),i−kDg−g1;m(B),k = 0

(using the convention that D0;∅,0 = 1/2).

This recursion relation determines Dg;m,i because all terms except the one with B = ∅,g1 = g involve lower genus or shorter partitions. In terms of the generating functions Dm,i

our recursion can be written∑

B⊆[l]

∑

k

(−1)k+|m(B)|dl+1−|B|Dm(B),i−kd|B|Dm(B),k = 0,

(for |m| < i− 1) where dj is short for dj

dxj .

Conjecture 4.3.1. The hyperelliptic Hodge integrals∫

M0;2g+2,0(BZ2)λgλg−iψ

m

are given by∑

g

(∫λgλg−iψ

m

)x2g

(2g)!=

(i− 1

m1 · · · ml

)2i−1

i!

(ln sec

x

2

)i

(the leading coefficient on the RHS is a multinomial coefficient).

Since we worked out a recursion determining these hyperelliptic Hodge integrals, all wehave to do is check that our conjectural solution satisfies the recursion. That is, we must


show that∑

B⊆[l]

∑

k

(−1)k+|m(B)| 2i−k−12k−1

i!(i− k)!

(k − 1m(B)

)(i− k − 1m(B)

)dl+1−|B|f i−kd|B|fk = 0,

where f = ln sec(x/2) and the terms with m(B) and m(B) are interpreted as multinomialcoefficients, so that, for example:(

k − 1m(B)

)=

(k − 1

mb1 · · · mbs k − 1− |m(B)|)

The fact that f = ln sec(x/2) is probably irrelevant. This identity should hold when f isany element of a ring equipped with a Z-linear operator d satisfying the Leibnitz Rule.Multiplying by a constant, we need to show

∑

B⊆[l]

∑

k

(−1)k+|m(B)|(ik

) (k − 1m(B)

)(i− k − 1m(B)

)dl+1−|B|f i−kd|B|fk = 0.

I have checked this identity for l ≤ 4.


5. The CRC in Degree Zero

In this section we prove that the Crepant Resolution Conjecture of Bryan and Graberholds “in degree zero” for involutions. We use the notation of Section 3: M is a smoothprojective variety with an involution, codimension 2 fixed locus C of dimension d, andquotient X. Let E denote the exceptional divisor of the crepant resolution r : Y → X.

The phrase “in degree zero” is in quotes because the CRC does not predict that the de-gree zero invariants of the stack X = [M/Z2] are determined by the degree zero invariantsof the resolution r : Y → X. This is because the quantum parameter qF corresponding tothe homology class of a fiber F of r over a point of C must be set to −1 in the potentialfunction of Y , and hence the invariants for homology classes e[F ] will contribute to thepotential of Y even when the other quantum parameters are set to zero. What we prove isthat the degree zero part of the Gromov-Witten theory of X is determined by the appro-priate part of the Gromov-Witten theory of Y as predicted by the CRC. The fact that the3-point invariants correctly coincide was proved by Perroni in his PhD thesis [Perroni]. Infact, he proved this for all orbifolds with transversal A1 or A2 singularities, not just globalquotients. Apparently this seriously complicates things because our proof is a lot simplerthan his. We are most indebted to Perroni for suggesting the correct change of variables.

We first explain how to compute the degree zero invariants of the orbifold X = [M/Z2].For g ≥ 0, we have

M0;2g+2,n(X , 0) = C ×M0;2g+2,n(BZ2)because the ramification points must map to the fixed locus, hence the degree zero mapjust collapses the domain onto a point of C. The twisted stable map spaceM0;2g+2,n(X , 0)has

Expected dimension : d− 1 + nActual dimension : d− 1 + n+ 2gObstruction class : c2g(NC/M £ E∨)

= c1(NC/M )c2g−1(E∨ ⊕ E∨) + c2(NC/M )c2g−2(E∨ ⊕ E∨)

The GW invariant 〈β〉0 vanishes by the Dimension Axiom unless β has degree d− 1 + n.It follows that all the invariants with n > 0, g > 0 vanish because C has dimension dso too much of the integrand is pulled back from the first factor of the product C ×M0;2g+2,n(BZ2). When n = 0, g > 0, 〈β〉0 vanishes unless the degree of

β = β1 ⊗ · · · ⊗ β2g+2 ∈ H∗(C)⊗(2g+2)

is d− 1, in which case we have

〈β〉0 =

(∫

Cc1(NC/M ) ·

∏

i

βi

)(−

∫

M0;2g+2,n(BZ2)λgλg−1

).

When g = 0, then by the Point Mapping Axiom (and stability), the only invariants thatcan be nonzero are of the form 〈β1, β2, α〉 with βi ∈ H∗(C) ⊂ H∗

CR(X ) and α ∈ H∗(X) ⊂H∗

CR(X ). These are given by

〈β1, β2, α〉 =12

∫

Cβ1β2α|C

because M0;2,1(X , 0) = BZ2×C. Notice that the degree zero GW theory of X does notdepend on c2(NC/M ).


Now we explain how to compute the GW invariants of the resolution Y for the homologyclass e[F ], where F ∼= P1 is a fiber of r : Y → X over a point of C ⊂ X. This is actuallyeasier to do when e > 0, which we now assume. Since F is collapsed by r, the maps inhomology class e[F ] are just multiple covers of fibers of r over C. The universal map f inthe diagram

Cf //

π²²

E ⊂ Y

M0;n(Y, e[F ])

hence factors through the exceptional divisor E so M0;n(Y, e[F ]) is the total space of afiber bundle

M0;n(P1, e) // M0;n(Y, e[F ])

w

²²C

and hasExpected dimension : d− 1 + nActual dimension : 2e− 2 + d+ nObstruction class : c2e−1(R1π∗f∗TY )

Note that the obstruction class is as indicated because the moduli space M0;n(Y, e[F ]) issmooth and R1π∗f∗TY is a vector bundle with rank equal to the excess dimension 2d− 1,as we shall now prove.

Since the universal map f factors through E, we have f∗TY = f∗(TY |E). Consider theSES

0 → TE → TY |E → NE/Y → 0

of vector bundles on E and apply R•π∗f∗. I claim that this gives an isomorphismR1π∗f∗TY = R1π∗f∗NE/Y because f∗TE has no higher cohomology on fibers. Indeed,consider the SES

0 → Tp → TE → p∗TC → 0

on E, with p : E → C the projection and Tp its relative tangent sheaf. The sheaf Tp ispositive on fibers of p and p∗TC is trivial on fibers of p so we have

R1π∗f∗Tp = R1π∗f∗p∗TC = 0

and hence R1π∗f∗TE = 0 as claimed.

Next recall that NE/Y = OE(−2) and that E is the projectivization of NC/M . Tensorthe Euler sequence

0 → OE(−1) → p∗NC/M → p∗NC/M/OE(−1) → 0

on E with OE(−1) to get a SES

0 → OE(−2) → p∗NC/M ⊗ OE(−1) → ∧2p∗NC/M → 0.

Here we have used thatA⊗ (V/A) → ∧2Va⊗ [v] 7→ a ∧ v


is a natural isomorphism of vector spaces (or free modules or locally free sheaves) when Ais a rank 1 subspace of a rank 2 vector space V . Taking R•π∗f∗ gives a SES

0 → π∗f∗p∗ ∧2 NC/M → R1π∗f∗OE(−2) → R1π∗f∗OE(−1)⊗ p∗NC/M → 0

so we havec2e−1(R1π∗f∗OE(−2)) = c1(π∗f∗p∗ ∧2 NC/M )c2e−2(R1π∗f∗OE(−1)⊗ p∗NC/M )

= w∗c1(NC/M )c2e−2(R1π∗f∗(OE(−1)⊗ p∗NC/M )).

To compute the GW invariant 〈γ〉e[F ], notice that the evaluation maps

ei : M0;n(Y, e[F ]) → Y

factor through E ⊂ Y . (We continue to assume e > 0.) By Leray-Hirsch we have

H∗(E) = H∗(C)[h]/〈h2 + c1(NC/M )h+ c2(NC/M )〉with h = c1(OE(1)), so we can write any cohomology class in H∗(E) uniquely in the formδ+βh for some δ, β ∈ H∗(C). In particular, for γ = γ1⊗· · · γn, we can write γi = δi +βih.To evaluate∫

M0;n(Y,e[F ])w∗c1(NC/M )c2e−2(R1π∗f∗(OE(−1)⊗ p∗NC/M ))

∏

i

e∗i γi

we first integrate over the fiber of w (that is, we take w∗). Notice that pei = w, so manyof these cohomology classes are pulled back from C via w and we can calculate w∗ bythe projection formula. The class h restricts to OP1(1) on fibers of E → C so w∗ of theintegrand is

c1(NC/M )β1 · · ·βn

(∫

M0;n(P1,e)c2e−2(R1π∗O(−1)⊕O(−1))

∏

i

e∗ih

)∈ H∗(C).

The term inside the parentheses is en−3 by the Aspinwall-Morrison formula and the DivisorAxiom for the GW theory of P1, so the GW invariant 〈γ〉e[F ] is given by

〈α〉e[F ] = en−3

∫

Cc1(NC/M )β1 · · ·βn.

Now we want to give the change of variables, compute the degree zero invariants of Y ,and show that the invariants match up correctly. The next two lemmas are due to Perroni.Cohomology groups here are with coefficients in Q(

√−1), as this is necessary to definethe correct change of variables. Consider the commutative diagram below.

E

π

²²

Â Ä j // Y

r

²²C

Â Ä

i// X

Lemma 5.0.2. For every i there is a split SES

0 // H i(X) r∗ // H i(Y ) // H i(E)/π∗H i(C) // 0

where the right map is the composition of j∗ and the cokernel map for π∗. The splittingis given by r∗ : H i(Y ) → H i(X).


Proof. Consider the long exact cohomology sequences for the pairs (Y,E) and (X,C):

· · · −−−−→ H i(Y,E) −−−−→ H i(Y )j∗−−−−→ H i(E) −−−−→ H i+1(Y,E) −−−−→ · · ·x

xr∗ π∗x

x· · · −−−−→ H i(X,C) −−−−→ H i(X) −−−−→

i∗H i(C) −−−−→ H i+1(X,C) −−−−→ · · ·

The cohomology of the pair (Y,E) is isomorphic to the cohomology of the pair (X,C)because the quotient topological spaces Y/E and X/C are homeomorphic (see Prop. 2.22in [40]), hence the arrows on the left and right are isomorphisms and j∗ induces an iso-morphism from the cokernel of r∗ to the cokernel of π∗. The splitting follows becauser∗r∗ = Id. Note that r∗ is defined using Poincare duality for X, which holds as long as 2is invertible in the coefficient ring. ¤

Corollary 5.0.3. The map

Ψ : H i(X)⊕H i−2(C) → H i(Y )(α, β) 7→ r∗α−

√−12 j∗π∗β

is an isomorphism of graded Q(√−1) vector spaces restricting to the identity on H i(X).

Proof. Make the identification H i(Y ) = H i(X) ⊕ (H i(E)/π∗H i(C)) using the splittingof Lemma 5.0.2. By Leray-Hirsch we can take h · π∗H i−2(C) as an additive basis forH i(E)/π∗H i(C). These facts together imply that the domain and codomain of Ψ areisomorphic as graded vector spaces, so we need only show that Ψ is surjective. First I claimthat any element of the form (α, 0) ∈ H i(Y ) is in the image of Ψ. Indeed, Ψ(α, 0) = (α, 0)because j∗(r∗α) = π∗i∗α ∈ π∗H i(C). This also proves that Ψ restricts to the identity onH∗(X). We can complete the proof by showing that for any β ∈ H i−2(C), there is someclass of the form (α, hπ∗β) in the image of Ψ. Indeed,

Ψ(0,−√−1β) = (√−1

2i∗β, hπ∗β)

because r∗j∗π∗ = i∗ and j∗j∗ is multiplication by the Euler class of the normal bundleNE/Y , which is −2h = c1(OE(−2)). ¤

The signs in our change of variables are inserted to arrange that, for β ∈ H i−2(C) ⊂H i

orb(X ), we have (Ψβ)|E =√−1hπ∗β ∈ H i(E).

Theorem 5.0.4. The genus zero, degree zero Gromov-Witten potential function for X isdetermined from the genus zero potential function for Y in homology classes e[F ] underthe change of variables Ψ of Corollary 5.0.3 as in the Crepant Resolution Conjecture 2.6.1.

The rest of this section is devoted to proving this theorem, which is now mostly a com-putational matter. Choose bases β0 = 1, β1, . . . , βr for H∗(C) and α1 = 1, α2, . . . , αs forH∗(X) so the αi and βi together form a basis for H∗

orb(X ). Then the degree zero potentialfunction ΦX is a formal power series in corresponding formal variables x0, . . . , xr+s.

First we show that all invariants involving α (untwisted) insertions match up. ForX , the only such invariants are of the form 〈αi, αj , αk〉X0 or 〈βi, βj , αk〉X0 . For Y , wehave already seen that the invariants for homology classes e[F ] with e > 0 vanish if anyinsertion γ has γ|E ∈ π∗H∗(C). But (Ψαi)|E = (r∗αi)|E = π∗(αi|C) so the invariants


〈Ψαi,Ψαj ,Ψαk〉Ye[F ] and 〈Ψβi,Ψβj ,Ψαk〉Ye[F ] can be nonzero only when e = 0. Certainlythe invariants with only α insertions match up because both are just given by integratingαiαjαk over X. For the other invariants, just note that j∗ : H∗(E) → H∗+2(Y ) ischaracterized by the property

(j∗γ) a [Y ] = j∗(γ a [E])

so we compute

〈Ψβi,Ψβj ,Ψαk〉Y0 =∫

Y(−√−1

2j∗π∗βi)(−

√−12

j∗π∗βj)r∗αk

= −14

∫

Eβi(j∗j∗π∗βj)j∗r∗αk

= −14

∫

Eβi(−2hβj)π∗(αk|C)

=12

∫

Cβiβj(αk|C)

=∫

C×BZ2

βiβj(αk|C)

= 〈βi, βj , αk〉X0as desired.

Now we only need to consider the invariants with only β insertions, that is, invari-ants with insertions only from the twisted sector H∗(C) ⊂ H∗

orb(X ). The invariants〈βi, βj , βk〉X0 with exactly three insertions play a special role because

〈Ψβi,Ψβj ,Ψβk〉Y0 = −√−13

8

∫

Y(j∗π∗βi)(j∗π∗βj)(j∗π∗βk)

=√−1

8

∫

Eβi(−2hβj)(−2hβk)

=√−1

2

∫

E(−c1(NC/M )h− c2(NC/M ))βiβjβk

= −√−1

2

∫

Cc1(NC/M )βiβjβk

but the degree 0 invariant for Y vanishes when there are more than three insertions.

By differentiating the expression for D1 obtained in Section 4, we get

∑

g≥1

(∫

M0;2g+2,0

λgλg−1

)x2g−1

(2g − 1)!=

12

tan(x/2).

It is convenient to consider all the GW invariants which differ only in the number of β0

insertions at the same time. Using our calculation of the GW invariants for X from above,we get

∑g≥0

2g+2≥n

〈βi1 , . . . , βin , β⊗2g+2−n0 〉X0

x2g+2−n

(2g + 2− n)!


=∑g≥0

2g+2≥n

(∫

Cβi1 · · ·βinc1(NC/M )

) (−

∫

M0;2g+2,0

λgλg−1

)x2g+2−n

(2g + 2− n)!

=(∫


)(d

dx

)n−3 (−1

2tan(x/2)

)

for any i1, . . . , in ∈ 0, . . . , r with n ≥ 3. This power series in x is the coefficient ofxi1 · · ·xin in the degree zero potential function of X (with x0 set to x) up to a factor ofAut(i1, . . . , in), at least when none of the ij are zero. If N of the ij are zero, then the samestatement holds, except we are instead considering the degree zero potential differentiatedN times with respect to x0.

Since (Ψβi)|E =√−1hβi, we can use our calculation of the GW invariants for Y from

above to get∑

d≥0

∑g≥0

2g+2≥n

〈Ψβi1 , . . . ,Ψβin ,Ψβ⊗2g+2−n0 〉Yd[F ]

x2g+2−n

(2g + 2− n)!qd

=(∫


)−

√−12

+∑g≥0

2g+2≥n

∞∑

d=1

√−12g+2

d2g+2−3 x2g+2−n

(2g + 2− n)!qd

=(∫


)−

√−12

+∞∑

d=1

√−1ndn−3

∑g≥0

2g+2≥n

(d√−1x)2g+2−n

(2g + 2− n)!

qd

=(∫


) (d

dx

)n−3(−√−1

2+√−1

∞∑

d=1

ed√−1xqd

)

The expressions −√−1

2 should be missing in the middle two lines if n > 3, but the finalequality is valid for all n ≥ 3. This power series in x, q gives a coefficient in the “degreezero” potential for Y corresponding to the coefficient in the degree zero potential for Xin the previous paragraph.

We can complete the proof by showing that

−√−1

2+√−1

∞∑

d=1

ed√−1xqd

admits an analytic continuation to q = −1 making it equal to −12 tan

(x2

). This is the same

analytic continuation used by Bryan and Graber in [14]. We repeat their calculation:

−√−1

2+√−1

∞∑

d=1

ed√−1x(−1)d =

12√−1

(1− e

√−1x

1 + e√−1x

)

= −12

tan(x

2

)

Note that in certain cases we have only proved equality of potential functions afterdifferentiating. This corresponds exactly to dropping unstable terms. Considering onlythe 3-point degree zero invariants, we recover Perroni’s isomorphism of cohomology rings:


Corollary 5.0.5. The orbifold cohomology ring H∗orb(X ) agrees with the cohomology ring

H∗(Y ) of the crepant resolution after replacing the structure constants for H∗(Y ) withtheir quantum deformations at q = −1, that is, using the structure constants:

∫

Yα1 · α2 · α3 :=

∑

d≥0

〈α1, α2, α3〉d[F ]Y qd

∣∣∣∣∣∣q=−1


6. Local Curves

In [17], Bryan and Pandharipande computed the local Gromov-Witten theory (in allgenera) of a curve in a threefold. This is the Gromov-Witten theory, defined by formalvirtual localization using the natural torus action, of the total space of a split rank twovector bundle on a curve. Roughly speaking, this is accomplished by degeneration ar-guments, with the result expressed in a TQFT formalism. Cavalieri’s PhD thesis [20]contains similiar results for moduli spaces of admissible covers. It would be nice to havean analogous result in orbifold Gromov-Witten theory for split rank two bundles L1 ⊕L2

on a curve C with, say, Zn-action (the action should be multiplication by a primitive nth

root of unity ζ on L1 and by ζ−1 on L2 to ensure the Gorenstein condition).

The orbifold theory poses new challenges. For one thing, the relative Gromov-Wittentheory developed by J. Li in [45] would need to be generalized to the orbifold case. Pre-sumably this is not terribly difficult. Secondly, the Divisor Axiom, which makes thecomputation of invariants with primary insertions unnecessary in the non-orbifold case, isunavailable for orbifolds. Indeed, we will be quite interested in the GW invariants withinsertions from the twisted sector.

Here we will consider the genus zero theory of the Z2-action on the total space M ofthe vector bundle

L1 ⊕ L2 = O(l1)⊕O(l2)

over P1. We being by establishing a WDVV-type recursion which can be used to recursivelyreduce the genus zero theory of X = [M/Z2] to invariants with one (nontrivial) insertionfrom the twisted sector. When (l1, l2) = (0,−1) we can explicitly compute the necessary“initial conditions” and hence determine the genus zero potential. It is given as the solutionto a simple differential equation.

The local theory described here is a nice testing ground for Bryan and Graber’s CrepantResolution Conjecture [14] (Conjecture 2.6.1). Indeed, M/Z2 is a locally trivial bundleof A1 singularities with a natural crepant resolution given by the total space of the linebundle OZ(−2) on the space Z := P(L1 ⊕ L2). In the (0,−1) case over P1 mentionedabove, we work out the genus zero potential of Y and verify the CRC in Section 6.4.

6.1. The Orbifold X . Let M be the total space of L1 ⊕ L2 = O(l1) ⊕ O(l2) over P1

and let X = [M/Z2], where Z2 acts by multiplication by −1 on L1 and L2. The 2-torusT = (C∗)2 acts on M by scaling in the Li. We identify H∗

T (pt) with Q[t1, t2] in the usualway. Note that the T fixed locus of X is the zero section, which we denote P1 × BZ2

throughout.

We fix a basis X0, X1, X2, X3 for the Chen-Ruan orbifold cohomology ring

H∗orb(X ) = H∗(X )⊕H∗−2(MZ2) = H∗(P1)⊕H∗−2(P1)

as in the table below.

Grading/Sector Untwisted Twisted0 X0 = 12 X1 = h X2 = 14 X3 = h


The expected dimension of the twisted stable map space M0;2g+2,n(X , d) is

n+ d〈c1(TX ), [P1 ×BZ2]〉 = n+ d(2 + l1 + l2).

The torus fixed locus M0;2g+2,n(X , d)T consists of maps onto the zero section, so we willidentify it with

Md,g,n := M0;2g+2,n(P1 ×BZ2, d),

which is a smooth stack of dimension 2d+ 2g + n (we will drop the subscript n when itis zero). In the following discussion we assume that there are marked ramification points(g ≥ 0). Over Md,g,n we have the universal base curve C, base map f , Z2 branched coverC, and equivariant map f :

C

π

ÀÀ

f //

q

²²

P1 Â Ä // M

²²C

π

²²

f // P1 Â Ä // M/Z2

Md,g,n

The upper part of the diagram specifies the universal twisted curve π : [C/Z2] → Md,g,n

and universal mapf : [C/Z2] → P1 ×BZ2 → X .

The virtual normal bundle of Md,g,n in M0;2g+2,n(X , d) is the element

Nvir = π!f∗(L1 ⊕ L2) = πZ2∗ f

∗(L1 ⊕ L2)ªR1πZ2∗ f∗(L1 ⊕ L2)

in the T -equivariant K-theory of Md,g,n. (The T action is trivial on Md,g,n.) We will callits top equivariant Chern class

Obg,n := c2g−d(l1+l2)(ªNvir) ∈ H∗T (Md,g,n)t1t2 = H∗(Md,g,n)[t1, t−1

1 , t2, t−12 ]

the obstruction class. It is the product of obstruction classes Ob1d,g,Ob2

d,g assigned to theline bundles L1, L2.

Since Z2 acts on Li by −1, the equivariant pushforward Rj πZ2∗ f∗L1 ⊕ L2 is just theanti-invariant part of Rj π∗f∗L1 ⊕ L2. Since the invariant part is Rjπ∗f∗L1 ⊕ L2, we canwrite the virtual normal bundle as

π∗f∗(L1 ⊕ L2)ªR1π∗f∗(L1 ⊕ L2)ª π∗f∗(L1 ⊕ L2)⊕R1π∗f∗(L1 ⊕ L2).

The individual terms here are not locally free on Md,g,n (though we can still take theChern classes since Md,g,n is smooth), though each term will be locally free on the locusin Md,g,n consisting of maps with smooth domain. Certainly if li < 0 then we only havethe higher direct images. In this case we have

Obid,g =

cg−2dli−1(R1π∗f∗Li)c−dli−1(R1π∗f∗Li)

∈ H2(g−dli)T (Md,g,n)t1t2 .

When li > 0, then at least f∗Li is a hyperelliptic line bundle on the (smooth) fibersof π and so H0 on such a fiber is as large as possible (c.f. Clifford’s Theorem [2]). (This


bound can be violated when the domain curve is not smooth.) On the locus of maps withsmooth domain, Grauert’s criterion implies that π∗f∗Li is locally free of rank

1 + dli, 0 ≤ dli ≤ g − 11− g + 2dli, g ≤ dli

and R1π∗f∗Li is locally free of rankg − dli, 0 ≤ dli ≤ g − 10, g ≤ dli.

When the degree is small relative to the genus (i.e. in the range 0 ≤ dli ≤ g − 1), thenatural inclusion

π∗f∗Li → π∗f∗Li

is an isomorphism and we have

Obid,g =

cg−dli(R1π∗f∗Li)c1+dli(π∗f

∗Li)c1+dli(π∗f∗Li)

= cg−dli(R1π∗f∗Li).

When the degree is large (dli ≥ g) we have

Obid,g =

c1+dli(π∗f∗Li)

c1+2dli−g(π∗f∗Li).

Notice that the obstruction class Obd,g has nothing to do with the number of unramifiedmarked points n. Indeed, the obstruction class on Md,g,n is pulled back from Md,g. Therank of the virtual normal bundle plus the dimension of the torus fixed locus is equal tothe expected dimension. Indeed, all obstructions are “moving” (have nonzero T weight)and the virtual fundamental class of the T -fixed locus Md,g,n is just its usual fundamentalclass. (Compare with Section 4 of [34].)

Example 6.1.1. Although the bundles π∗f∗Li and R1π∗f∗Li have constant rank on thelocus of curves with smooth domain, this need not be true on nodal curves. For example,in M0;2g+2,0([(O⊕O(1))/Z2], 1), a general domain curve C is a smooth hyperelliptic curveof genus g and f∗O(1) is hyperelliptic of degree 2 with a 2-dimensional space of globalsections (all pulled back from P1) and (g − 1)-dimensional H1. However, if C consists ofa genus zero component glued to a collapsed genus g component, then H0 and H1 havedimensions 3 and g respectively.

For(α, β) ∈ H∗

T (X )⊗n ⊕H∗T (MZ

2 )⊗2g+2,

the virtual localization formula [34] expresses the Gromov-Witten invariant 〈α, β〉d of Xin terms of an integral over the torus fixed locus in the space of stable maps to X :

〈α, β〉d =∫

[M0;2g+2,n(X ,d)]vir

e∗α ` e∗β

=∫

[M0;2g+2,n(X ,d)T ]vir

e∗α ` e∗βe(Nvir)

=∫

Md,g,n

Obd,g ` e∗α ` e∗β ∈ Q[t1, t−11 , t2, t

−12 ]


The moduli spaceM0;2g+2,n(X , d) need not be compact, so the first expression may not bewell-defined, but, as usual, the third expression is certainly well-defined, so we simply takeit as the definition of the GW invariant. The integrals of course mean equivariant pushfor-ward to a point; they take values in H∗

T (pt)t1t2 , which we identify with Q[t1, t−11 , t2, t

−12 ].

We will decorate the GW invariant with a subscript 〈α, β〉l1,l2 if the line bundles L1 ⊕ L2

are not clear from context. Note that the evaluation maps

ei : M0;2g+2,n(X , d) → X i = 1, . . . , n

restricted to the torus fixed locus Md,g,n naturally take values in X T = P1 × BZ2. Wewill usually just compose with the projection to the coarse space and view them as takingvalues in P1.

Recall from 2.5 that we assemble these invariants into a potential function

ΦX =∑

d

∑n0,n1,n2,n3

〈X⊗n00 , X⊗n1

1 , X⊗n22 , X⊗n3

3 〉dxn00 xn1

1 xn22 xn3

3

n1!n2!n3!n4!qd.

The degree zero stable map spaces with three marked points are

M0;2,1(X , d) = M0;2,1(X , d)T = P1 ×BZ2 and M0;0,3(X , 0) = X

as usual. We have M0;0,3(X , 0)T = P1 and the “virtual normal bundle” of M0;0,3(X , 0)T

in M0;0,3(X , 0) is just its usual normal bundle L1⊕L2. The 3-point degree zero invariantsare easily computed. (There are factors of 2 due to automorphisms.)

Lemma 6.1.1.〈X0, X2, X2〉0 = 0 〈X0, X0, X0〉0 = −1

2(l1t−21 t−1

2 + l2t−11 t−2

2 )〈X0, X2, X3〉0 = 1

2 〈X0, X0, X1〉0 = 12 t−11 t−1

2〈X0, X3, X3〉0 = 0 〈X0, X1, X1〉0 = 0〈X1, X2, X2〉0 = 1

2 〈X1, X1, X1〉0 = 0〈X1, X2, X3〉0 = 0〈X1, X3, X3〉0 = 0

The (formal equivariant) orbifold Poincare pairing gX = 〈[X ], XiXj〉ij and its inverseare given below.

gX =

0 12 t−11 t−1

2 0 012 t−11 t−1

2 0 0 00 0 0 1

20 0 1

2 0

g−1

X =

0 2t1t2 0 02t1t2 0 0 0

0 0 0 20 0 2 0

For higher point or positive degree invariants, X0, X1 insertions can be removed withthe Point Mapping and Divisor Axioms, so invariants with only untwisted insertions arereduced to invariants with no insertions. In some cases these have nice expressions:

Lemma 6.1.2.

〈〉d0,0 =

12 t−11 t−1

20

d = 1d > 1

〈〉d0,−1 =

12 t−11

0d = 1d > 1

〈〉d−1,−1 = 12d−3 d > 0

〈〉d0,−2 = 12 t−11 t2d

−3 d > 0


Proof. When there are no marked ramification points (so g = −1 in the above discussion),the universal curve

π : C →M0;0;0(P1 ×BZ2, d)is disconnected. It is just the disjoint union of two copies of the universal curve

π : C →M0,0(P1, d).

The bundles Rjπ∗f∗Li are just given by taking the anti-diagonal of

Rj π∗f∗Li = Rjπ∗f∗Li ⊕Rjπ∗f∗Li.

Depending on whether li is positive or negative, the obstruction class is either

Obd,−1 = c1+dli(π∗f∗Li) or c−dli−1(R1π∗f∗Li).

The desired invariants are now computed by the Aspinwall-Morrison formula and theusual trick relating the penultimate Chern class of R1π∗f∗O(−2) to the Euler class ofR1π∗f∗(O(−1)⊕O(−1)). There are factors of 2 because of the automorphism exchangingthe two copies of C. ¤Remark 6.1.1. The invariants 〈〉dl1,l2

can be determined for arbitrary l1, l2 by the recur-sions of [14], but I am not aware of any nice formulas.

As in Section 5, the degree zero invariants are also easy to compute:

Lemma 6.1.3. The degree zero invariants 〈X⊗s3 , X⊗2g+2−s

2 〉0 are given by

−(l1 + l2)∫

M0;2g+2,0(BZ2)λgλg−1 when s = 0

−(t1 + t2)∫

M0;2g+2,0(BZ2)λgλg−1 when s = 1

and vanish for s > 1. (Generating series for these integrals appear in Section 4.)

Proof. When d = 0 we have

M0;2g+2,0(X , 0) = P1 ×M0;2g+2,0(BZ2)

with the obstruction class given by

c2g((L1 ⊕ L2) £ E∨),

where E is the Hodge bundle on M0;2g+2,0(BZ2) (see Section 4). The lemma follows byexpanding the Chern class and collecting the terms of the correct degree to integrate overthe product. ¤

6.2. WDVV. In this section we show that an appropriately-modified version of the usualWDVV formula holds for the formal equivariant genus zero potential function of X . Weuse this to give a recursion reducing arbitrary (genus zero, primary) GW invariants of Xto those with at most one X3 insertion, plus the two point invariant 〈X3, X3〉dl1,l2

.

Define the modified inverse orbifold Poincare pairing gX = (gabX ) as below.

gX =

0 2t1t2 0 02t1t2 2(l1t2 + l2t1) 0 0

0 0 0 20 0 2 0


This differs from the usual inverse pairing matrix gabX only in the (1, 1) entry.

Proposition 6.2.1. The usual WDVV equation∑

a,b

gabX ΦX

i,j,aΦXk,l,b =

∑

a,b

gabX ΦX

i,k,aΦXj,l,b

holds for the formal equivariant genus zero potential function ΦX for any i, j, k, l ∈0, 1, 2, 3. Here gab

X is the (a, b) entry in the modified inverse orbifold Poincare pair-ing matrix defined above. The subscripts are partial derivatives with respect to the xi

variables.

Proof. Consider the moduli space Md,g,n = M0;2g+2,n(X , d)T . The WDVV equation isproved by considering two linearly equivalent boundary divisorsD1, D2 onMd,g,n where theuniversal domain curves are nodal and the marked points are split (i, j|k, l) and (i, k|j, l),respectively. (That is, WDVV is obtained by pulling back a relation on M0,4

∼= P1 via theforgetful stabilization maps Md,g,n → M0,4 which retain only the genus zero base curveand four markings.) Each of the divisors Di is expressed (up to an automorphism of thegluing) as a disjoint union of divisors D defined by one of the cartesian diagrams below.

Md,g,n D?_ioo

eP=Q

²²

Â Ä j // Md1,g1,n1+P ×Md2,g2,n2+Q

e=eP×eQ

²²P1 ×BZ2

Â Ä ∆ // (P1 ×BZ2)× (P1 ×BZ2)

Md,g,n D?_ioo

eP=Q

²²

Â Ä j // Md1,g1,n1 ×Md2,g2,n2

e=eP×eQ

²²P1 ×BZ2

Â Ä ∆ // (P1 ×BZ2)× (P1 ×BZ2)

In the top diagram, the gluing is at unramified marked points P,Q and we have

g1 + g2 = g − 1, n1 + n2 = n, d1 + d2 = d.

In the bottom diagram, the gluing is at ramified marked points P,Q and we have

g1 + g2 = g, n1 + n2 = n, d1 + d2 = d.

I claim that on a component of the former type, we have

i∗Nvird,g ⊕ e∗P=Q(L1 ⊕ L2) = j∗(Nvir

d1,g1¢Nvir

d2,g2)

while on a component of the latter type the virtual normal bundles pulled back via i andj coincide.

Let us consider the first case. Write M1,M2 for Md1,g1,n1+P ,Md2,g2,n2+Q and use thenotation πi : Ci → M1 for the universal covering curve on Mi and πi, fi, fi, etc. for theother universal objects. The normalization sequence for the universal Z2 cover C is

0 → OC → OC1⊕OC2

→ OD → 0


where D is the divisor of gluing nodes, which is a principal Z2-bundle over D (or rather,its image in the universal base curve under the unramified section P = Q). Tensoringwith f∗Li and taking π! we get an equality

π1!f∗1Li ⊕ π2!f

∗2Li = π!f

∗Li ⊕ e∗2(Li ⊕ Li)

in the T -equivariant K group KT (D). There are two factors of Li because the divisor ofnodes is a principal Z2 bundle.5 If we instead examine the normalization sequence for theuniversal base curve, then the divisor of gluing nodes is nothing but the image of D andwe get

π1!f∗1Li ⊕ π2!f

∗2Li = π!f

∗Li ⊕ e∗2Li.

Adding and subtracting terms in K-theory and taking Chern classes gives the desiredequality of virtual normal bundles. The computation for the case where we glue at aramified points proceeds in the same manner. In this case there is no extra factor of Li

because the divisor of nodes in the universal covering curve is isomorphic to D rather thana Z2-bundle over it.

The next point to make is that when the gluing is at unramified points, the evaluatione = eP ×eQ takes values in the untwisted sector X of IX and we can write the homologyclass of D in M1 ×M2 as

[D] = e∗(2X0 ⊗X1 + 2X1 ⊗X0) a [M1 ×M2].

To prove the WDVV equation, one considers a cohomology class α on Md,g,n pulled backby evaluation maps and integrates over the formal virtual class ofD. That is, one considers

∫

Di∗α

1i∗e(Nvir

d,g ).

Since α is pulled back by evaluation maps, i∗α can also be written as j∗ of some cohomologyclass, which we will also call α, pulled back from M1×M2 via the evaluation maps. Usingour formula for the class D inside of the product M1×M2 and our relation between virtualnormal bundles with the computation above, we can express the integral above as

∫

M1×M2

j∗αe∗(2X0 ⊗X1 + 2X1 ⊗X0)e∗P=Q(c2(L1 ⊕ L2))

j∗(Nvird1,g1

¢Nvird2,g2

).

The numerator of the fraction can be expanded as

2t1t2e∗(X0 ⊗X1) + 2(l1t2 + l2t1)e∗(X1 ⊗X1) + 2t1t2e∗(X1 ⊗X0).

When the gluing is at ramification points, the story is similar. This time the class of Din the product is

[D] = e∗(2X2 ⊗X3 + 2X3 ⊗X2) a [M1 ×M2]

and the virtual classes agree. The expressions derived for the class of D in M1×M2 (aftercorrecting for the difference in virtual normal bundles) are given by the usual formula

∑

a,b

gabX Xa ⊗Xb

(pulled back by the appropriate evaluation maps).

5If D is not a trivial Z2 principal bundle, then one of the two Li factors might be tensored by a line bundlewith trivial square, but this is irrelevant for our computations.


Now the usual proof of WDVV applies: to equate coefficients on the two sides of theWDVV equation, one simply chooses an appropriate α (depending on the coefficients onewishes to equate) and integrates it (formally) over D1 and D2. ¤

Scholium 6.2.1. The comparison of virtual normal bundles on divisors of nodal curvesshould be viewed as an equivariant version of the Splitting Axiom (aka Cutting EdgesAxiom) for the virtual fundamental class in Gromov-Witten theory. For example, whengluing along unramified sections, we considered the diagram below.

Md,g,n D?_ioo

eP=Q

²²

Â Ä j // Md1,g1,n1+P ×Md2,g2,n2+Q

e=eP×eQ

²²P1 ×BZ2

Â Ä ∆ // (P1 ×BZ2)× (P1 ×BZ2)

This is nothing but the T -fixed part of the diagram

W

e

²²

Â Ä // M0;2g1+2,n1+P (X , d1)×M0;2g2+2,n2+Q(X , d2)

eP×eQ

²²X

Â Ä ∆ // X ×X

defining a divisor W in M0;2g+2,n(X , d) with W T = D. The Splitting Axiom (see Defi-nition 7.1(3) in [16], Axiom III on Page 8 of [11], or Proposition 5.3.1 in [4] for orbifolds)then asserts that

i![M0;2g+2,n(X , d)]vir = ∆!([M0;2g1+2,n1+P (X , d1)]vir × [M0;2g2+2,n2+Q(X , d2)]vir

),

where i!,∆! are the refined Gysin homomorphisms of Fulton (see Chapter 6 of [30] or[54] for stacks). However, recall that in our situation we do not actually try to definethese virtual fundamental classes in general (for example because they might have negativeexpected dimension): we instead work with the formal localization sums. (We could definesuch virtual fundamental classes if, for example, (l1, l2) = (0,−1), in which case the stablemap spaces are compact of positive expected dimension.) Let us at least argue heuristicallythat our obstruction class expression is just an instance of the Splitting Axiom.

Assume that l1, l2 ≤ 0. Then the universal maps actually factor through the zero sectionso the vertical arrows in the above diagram factor through the cartesian square below.

P1 ×BZ2Â Ä∆P1×BZ2 //

²²

(P1 ×BZ2)× (P1 ×BZ2)

²²X

Â Ä ∆ // X ×X

The Excess Intersection Formula (9.3 in [30]) asserts that

∆!( ) = c2(e∗E) a ∆!P1×BZ2

( ),

where E is a certain excess intersection bundle on P1 × BZ2. Here E is nothing but thenormal bundle of the zero section in X , which is L1 ⊕ L2. The restriction of the virtualfundamental class to the torus fixed locus Md,g,n is Obd,g a [Md,g,1] (heuristically) so theexpression we just derived is the same as the one we arrived at by comparing obstructionbundles above.


A similar analysis applies when we glue at a ramified section, except now we form ourfibered diagrams with respect to evaluation maps taking values in the P1×BZ2 componentof the rigidified cyclotomic inertia stack IX = X

∐P1. In this case no excess intersection

formula is necessary.

It is fortunate that in the situation considered here, the T -fixed loci themselves havevirtual fundamental classes equal to their actual fundamental classes (all obstructions aremoving). In general, it should be possible, using the techniques of virtual localization[34], to compare virtual fundamental classes of the T -fixed loci and their virtual normalbundles and to formulate an appropriate equivariant version of the Splitting Axiom. Themethods used here are an ad hoc version of this.

Lemma 6.2.2. For any cohomology classes β1, . . . , βt from the twisted sector of H∗orb(X)

and any d ≥ 0, the relations

∑

i

∑

|V | odd

2〈X2, X2, V,X2〉i〈X1, X3, V, X3〉d−i

= t1t2〈X1, X2, [t], X1〉d +∑

i

∑

|V | odd

2〈X2, X3, V,X3〉i〈X1, X2, V, X2〉d−i

and

t1t2〈X1, X3, [t], X1〉d +∑

i

∑

|V | odd

2〈X3, X2, V,X2〉i〈X1, X3, V, X3〉d−i

=∑

i

∑

|V | odd

2〈X3, X3, V,X3〉i〈X1, X2, V, X2〉d−i

hold for the GW invariants of X . Here the sums are over all sets of insertions V ⊆β1, . . . , βt and V := β1, . . . , βt V .

Proof. Apply the WDVV equations (Proposition 6.2.1) with (i, j, k, l) equal to (2, 2, 1, 3)and (3, 2, 1, 3) to get the relations

∑d1+d2=d

∑|V | even 2〈X2, X2, V,X0〉d1〈X1, X3, V

, c2(L1 ⊕ L2)X1〉d2

+2〈X2, X2, V, c2(L1 ⊕ L2)X1〉d1〈X1, X3, V, X0〉d2∑

d1+d2=d

∑|V | odd 2〈X2, X2, V,X2〉d1〈X1, X3, V

, X3〉d2

+2〈X2, X2, V,X3〉d1〈X1, X3, V, X2〉d2

=∑

d1+d2=d

∑|V | even 2〈X2, X3, V,X0〉d1〈X1, X2, V

, c2(L1 ⊕ L2)X1〉d2

+2〈X2, X3, V, c2(L1 ⊕ L2)X1〉d1〈X1, X2, V, X0〉d2∑

d1+d2=d

∑|V | odd 2〈X2, X3, V,X2〉d1〈X1, X2, V

, X3〉d2

+2〈X2, X3, V,X3〉d1〈X1, X2, V, X2〉d2

and


∑d1+d2=d

∑|V | even 2〈X3, X2, V,X0〉d1〈X1, X3, V

, c2(L1 ⊕ L2)X1〉d2

+2〈X3, X2, V, c2(L1 ⊕ L2)X1〉d1〈X1, X3, V, X0〉d2∑

d1+d2=d

∑|V | odd 2〈X3, X2, V,X2〉d1〈X1, X3, V

, X3〉d2

+2〈X3, X2, V,X3〉d1〈X1, X3, V, X2〉d2

=∑

d1+d2=d

∑|V | even 2〈X3, X3, V,X0〉d1〈X1, X2, V

, c2(L1 ⊕ L2)X1〉d2

+2〈X3, X3, V, c2(L1 ⊕ L2)X1〉d1〈X1, X2, V, X0〉d2∑

d1+d2=d

∑|V | odd 2〈X3, X3, V,X2〉d1〈X1, X2, V

, X3〉d2

+2〈X3, X3, V,X3〉d1〈X1, X2, V, X2〉d2

(The sums are over V as in the statement of the lemma.)

These expressions can be simplified greatly by the Point Mapping and Divisor Axioms.In both cases, one of the terms is repeated on both sides of the equals sign, so we cancancel that too. ¤Proposition 6.2.3. The potential function ΦX is determined by the WDVV equations,two point invariants, degree zero invariants, and invariants with at most one X3 insertion.

Proof. This is a purely formal argument about the WDVV equation, which can be provedin much the same manner as Kontsevich’s Reconstruction Theorem. See Lemma 7.12.1.

¤

6.3. The case (l1, l2) = (0,−1). In this section we will derive explicit formulas for theGromov-Witten theory of X when (l1, l2) = (0,−1). In this case, the stable map spacesM0;2g+2,n(X , d) are compact as long as there are marked ramification points (g ≥ 0), sothe formal localization sum is actually an equivariant integral over a compact space. Theobstruction class Obd,g is given by

c2g+d

(R1π∗f∗(O ⊕O(−1))R1π∗f∗(O ⊕O(−1))

)= cg(R1π∗f∗O)cg+d

(R1π∗f∗O(−1)R1π∗f∗O(−1)

)

(the bundle f∗O is non-equivariantly trivial, but has weight t1).

There is no need to consider invariants with X0, or X1 insertions because of the DivisorAxiom, Point Mapping Axiom, and Lemma 6.1.2. The invariant 〈X⊗s

3 , X⊗2g+2−s2 〉d van-

ishes for dimension reasons unless s ≥ d. It now follows from Proposition 6.2.3 that thepotential ΦX is determined by the untwisted invariants of Lemma 6.1.2, the degree zeroinvariants of Lemma 6.1.3, and the invariants 〈X3, X

⊗2g+12 〉10,−1, which are computed in

the next lemma.

Lemma 6.3.1. The degree 1 invariants are given by the generating series∑

g≥0

〈X3, X⊗2g+12 〉10,−1

x2g+1

(2g + 1)!= −1

2tan

(x2

).

Proof. We computed the integral

〈X3, X⊗2g+12 〉10,−1 =

∫

M0;2g+2,0(P1×BZ2,1)e∗1(X3)c2g+1(R1π∗f∗(O ⊕ O(−1))

in the course of proving Recursion 2. in Section 4. ¤


We have now completely determined the potential function of [O ⊕ O(−1)/Z2]. Forlater use, we will derive explicit formulas for the “non-equivariant” GW invariants of Xby solving the WDVV recursions explicitly. That is, we wish to consider the invariants

Zd,g := 〈X⊗d3 , X⊗2g+2−d

2 〉d0,−1

which do not automatically vanish in the non-equivariant limit t1 = t2 = 0. We assemblethese into the generating series

Zd(x) :=∑

g

Zd,gx2g+2−d

(2g + 2− d)!

Z = Z(x, y) :=∑

d≥0

Zdyd

d!=

∑

d,g≥0

Zd,gx2g+2−d

(2g + 2− d)!yd

d!∈ Q[[x, y]].

Theorem 6.3.2. The function Z is determined by the second-order PDE in Zy

(Zy)yy = 4(Zy)y(Zy)xx

and the initial conditions below.

Zxx(x, 0) = ln sec(x

2

)

Z(0, 0) = 0Zx(0, 0) = 0

Zy(x, 0) = −12

tan(x

2

)

Zyy(x, 0) =116

sec4(x

2

)

Proof. All the initial conditions except the last are determined in Lemmas 6.1.2, 6.1.3,and 6.3.1. Applying the WDVV equation (Proposition 6.2.1) with (i, j, k, l) = (1, 1, 2, 3)and looking at the coefficient of x2g

2 , we get the relation2g∑

i=0

2(

2g2i+ 1

)〈X⊗i

2 , X3〉1〈X⊗2g+2−i2 , X3〉1 − 2t1〈〉1〈X⊗2g+1

2 , X3〉1

= 2〈X1, X2, X2〉0〈X1, X3, X⊗2g2 , X3〉2 + 2

2g∑

i=0

(2g2i

)〈X⊗2i+1

2 , X3〉1〈X⊗2g+1−i2 , X3〉1.

In terms of generating functions, we get

2Z1Z′′1 − Z1 = 2Z2 + 2(Z ′1)

2,

from which we can solve easily for Z2 = Zyy(x, 0) using the expression for Z1 fromLemma 6.3.1. (We could also have used the recursion in Lemma 6.2.2.)

Next apply the second WDVV equation in Lemma 6.2.2 with the tuple of cohomologyclasses

β = X3, . . . , X3︸︷︷︸d−3

, X2, . . . , X2︸︷︷︸2g+2−d

,

evaluate at t1 = t2 = 0, to find that for any d ≥ 3 and any g with 2g + 2 ≥ d we have:d−2∑

d1=1

(d− 3d1 − 1

)(∑g1

(2g + 2− d2g1 − d1

)(d− d1)Zd1,g1Zd−d1,g−g1

)


=12Zd,g +

d∑

d1=3

(d− 3d1 − 3

)(∑g1

(2g + 2− d

2g1 + 2− d1

)(d− d1)Zd1,g1Zd−d1,g−g1

)

This can be expressed in terms of the generating function Z to yield:12Zyyy = 2ZyyZxxy

(there are some fortunate cancellations). ¤

We summarize this section with

Theorem 6.3.3. The genus zero equivariant Gromov-Witten potential function of X isdetermined by the WDVV equations, its degree zero part:

ΦX |q=0 =12x0x2x3 +

112t−11 t−2

2 x30 +

14x1x

22 +

14t−11 t−1

2 x20x1

+H(x2)− (t1 + t2)H ′(x2)x3

and the terms in the non-equivariant evaluation t1 = t2 = 0 which are of degree one:

ΦX |t1=t2=0 = ΦX |q,t1,t2=0 − 12

tan(x2

2)x3e

x1q +O(q2).

Here H(x) is determined by H(0) = H ′(0) = H ′′(0) = 0 and H ′′′(x) = 12 tan

(x2

).

6.4. Resolutions of local curves. In this section we describe the crepant resolution Yof

X = [M/Z2] = [L1 ⊕ L2/Z2]

and begin computing its potential function. Let Z be the Hirzebruch surface

Z = P(L1 ⊕ L2)

and let Y be the total space of OZ(−2). Then Y inherits a natural torus action from theone on L1 ⊕ L2. Let p1 : Z → P1 and p2 : Y → Z be the natural projections.

Given any 2-dimensional vector space V , acted on by Z2 by multiplication by −1, andan element v1⊗v2 in the fiber of OPV (−2) over v ∈ PV (i.e. in v⊗v) we can write v2 = zv1for some z ∈ C. The map

v1 ⊗ v2 7→ [√zv1] ∈ V/Z2

is well-defined (independent of the choice of representation v1 ⊗ v2 and of the choice of√z) and yields a crepant resolution OPV (−2) → V/Z2 which is natural in V . This is the

same resolution obtained by blowing up as in Theorem 3.0.2. Since this map is natural inV we can do it on each fiber of the bundle of A1 singularities M/Z2 → P1 to get a map

r : Y →M/Z2.

This map is a T -equivariant crepant resolution of singularities commuting with the pro-jections to P1. The exceptional divisor of r is Z ⊂ Y and the curves collapsed by r arethe fibers of p1.

By Leray-Hirsch we have

H∗T (Y,Z) = H∗

T (Z,Z) =Z[g, h, t1, t2]

〈h2, g2 + (l1h+ t1 + l2h+ t2)g + t1t2 + l1t2h+ l2t1h〉 ,


where h is pulled back from P1 and g = c1(OZ(1)). Since p1 and p2 have sections, thetangent bundle of Y splits as

TY = p∗1TP1 ⊕ Tp1 ⊕ p∗2OZ(−2).

The two sectionsPL1,PL2 ⊂ Z = P(L1 ⊕ L2)

form the T -fixed locus of Y . We identify them with P1 by the projection p1 throughout.Their normal bundles (in Z) are Hom(L1, L2) and Hom(L2, L1) respectively. The normalbundle NPLi/Z agrees with the restriction of the relative tangent bundle Tp1 |PLi . The factthat OZ(−2)|PLi = Li ⊗ Li then completely describes the tangent bundle of Y on thesesections.

Since Y is not compact, some spaces of stable maps to Y are not compact, so we willdefine Gromov-Witten invariants by formal virtual localization as in Section 6. Note thatthe only T -invariant curves in Y are the sections PLi and the fibers of p1. The image ofany T -fixed stable map is a “ladder” of curves with the PLi forming the rails and finitelymany fibers forming the rungs (some pieces of the ladder may be missing of course). It ispossible to describe the fixed loci in terms of stable maps to P1, but the combinatorics isquite involved and we have no need to get into this.

We use the cohomology basis Y0 = 1, Y1 = h, Y2 = g, Y3 = gh throughout. As a basisfor H2(Y,Z), we can take the class of one of the sections, say PL1, and the class of a fiberof p1. We will denote the corresponding quantum parameters q1 and q2, respectively. Iconjecture that the potential functions ΦX (x0, x1, x2, x3, q) and ΦY (y0, y1, y2, y3, q1, q2) ofX and Y are related by the change of variables

ΦY (x0, x1, ix2, ix3,±q,−1) = ΦX (x0, x1, x2, x3, q).

(We must analytically continue to q2 = −1 and drop unstable terms as usual. See Conjec-ture 2.6.1.) The sign ambiguity ±q is resolved by demanding that the rigid section (theone with the most negative normal bundle O(−|l1 − l2|)) be taken to q. Any two sectionsdiffer in homology by some number of fibers and we insist that the fiber class be set to−1.

Lemma 6.4.1. The degree zero invariants of Y are given by

〈Y0, Y0, Y0〉0,0 = −12(l1t−2

1 t−12 + l2t

−11 t−2

2 ) 〈Y0, Y0, Y1〉0,0 = 12 t−11 t−1

2〈Y0, Y0, Y2〉0,0 = 0 〈Y0, Y0, Y3〉0,0 = 0〈Y0, Y1, Y2〉0,0 = 0 〈Y0, Y2, Y2〉0,0 = 0〈Y0, Y2, Y3〉0,0 = −1

2 〈Y1, Y2, Y2〉0,0 = −12

〈Y2, Y2, Y2〉0,0 = 12(l1 + l2) 〈Y2, Y2, Y3〉0,0 = 1

2(t1 + t2)

(Any such invariant where the number of Y1 and Y3 insertions exceeds one vanishes becauseh2 = 0.)

Proof. By their definition via formal virtual localization, such invariants are given by

〈Yi, Yj , Yk〉 =∫

PL1

Yi|PL1 ` Yj |PL1 ` Yk|PL1

c2(NPL1/Y )+

∫

PL2


c2(NPL2/Y )

=∫

PL1


c2((L∨1 ⊗ L2)⊕ L⊗21 )

+∫

PL2


c2((L∨2 ⊗ L1)⊕ L⊗22 )

.

Now we just calculate using the fact that g|PLi = c1(OZ(1)|PLi) = c1(L∨i ). ¤


The (formal equivariant) Poincare pairing gY = 〈YiYj , [Y ]〉 and its inverse are givenbelow.

gY =

0 12 t−11 t−1

2 0 012 t−11 t−1

2 0 0 00 0 0 −1

20 0 −1

2 0

g−1

Y =

0 2t1t2 0 02t1t2 0 0 0

0 0 0 −20 0 −2 0

Note that the change of variables suggested above respects the Poincare pairings on Xand Y .

Lemma 6.4.2. The invariants in homology class (0, d) (multiple covers of fibers) vanishwhen there is more than one X3 insertion. They are hence determined by the DivisorAxiom and the invariants

〈〉0,d = d−3(l1 + l2) (d > 0)〈Y3〉0,d = d−2(t1 + t2) (d > 0).

Proof. The relevant moduli spaceM0,n(Y, (0, d)) consists of multiple covers of fibers. Eachsuch map is fixed by the T action. The obstruction class is c2d−1(R1π∗f∗OZ(−2)). As inthe proof of Theorem 5.0.4 we consider the exact sequence

0 → OZ(−2) → p∗1(L1 ⊕ L2)⊗OZ(−1) → L1 ⊗ L2 → 0

to find

〈α〉0,d =∫

M0,n(Y,(0,d))e∗α ` c2d−2(p∗(L1 ⊕ L2)⊗OZ(−1))c1(L1 ⊗ L2).

Since M0,n(Y, (0, d)) fibers over the base P1 with fiber M0,n(P1, d) we can calculate thisintegral by pushing forward and using Aspinwall-Morrison and the Divisor Axiom to in-tegrate over the fibers. ¤

Proposition 6.4.3. The usual WDVV equation∑

a,b

gabY ΦY

i,j,aΦYk,l,a =

∑

a,b

gabY ΦY

i,k,aΦYj,l,a

holds for the equivariant genus zero potential of Y for any tuple (i, j, k, l) ∈ 0, 1, 2, 3using the modified inverse Poincare pairing matrix gY below.

gY =

0 2t1t2 0 02t1t2 2(l1t2 + l2t1) 0 0

0 0 0 −20 0 −2 0

These equations determine ΦY from the 2-point invariants.

Proof. We proceed as in the proof of Proposition 6.2.1. Fix a degree d = (d1, d2) withd2 > 0. Fix splittings n1 + n2 = n, d′ + d′′ = d of the marked points and degree and let

M1 := M0,n1+P (Y, d′), M2 := M0,n2+Q(Y, d′′).


Consider the cartesian diagram

M0,n(Y, d) D?_ioo Â Ä j //

eP=Q

²²

M1 ×M2

e=eP×eQ

²²Y

Â Ä ∆ // Y × Y

defining a boundary divisor D in M0,n(Y, d).

Consider the T -fixed part of this diagram:

M0,n(Y, d)T DT? _ioo Â Ä j //

eP=Q

²²

MT1 ×MT

2

e=eP×eQ

²²PLi

Â Ä ∆ // PLi × PLi

(We have fixed one of the components of D where eP=Q takes values in PLi.) Let Nvir bethe virtual normal bundle of M0,n(Y, d)T in M0,n(Y, d) and let Nvir

i be the virtual normalbundle of MT

i in Mi. The WDVV equation is obtained by considering a cohomology classα on M0,n(Y, d)T pulled back from Y by evaluation maps and looking at the integral

∫

[DT ]i∗α

1i∗e(Nvir)

.

We want to express this as an integral over MT1 ×MT

1 .

Examining the normalization sequence for the universal curve and considering movingparts of the tangent-obstruction complex, we find that

i∗Nvir ⊕ e∗P=QNPLi/Y = j∗(Nvir1 ¢Nvir

2 ).

The class of DT inside MT1 ×MT

2 is given by e∗(Y0⊗Y1 +Y1⊗Y0). A direct computationshows that

e∗(Y0 ⊗ Y1 + Y1 ⊗ Y0)e∗P=Q(c2(NPLi/Y )) =∑

a,b

gabY e

∗(Ya ⊗ Yb)

so it follows that the above integral can be written as∑

a,b

gabY

∫

MT1 ×MT

2

j∗αe∗(Ya ⊗ Yb)j∗Nvir

1 ¢Nvir2

.

The WDVV equation now follows from the usual formal argument.

¤

6.5. The resolution of (O ⊕O(−1))/Z2. Throughout this section, Z denotes the Hirze-bruch surface P(O ⊕ O(−1)) and Y is the total space of OZ(−1). The cone of effectivecurves in Z is spanned by the rigid section R := PO and the class of a fiber F of the projec-tion p1 : Z → P1. The same is true of Y , but note that OZ(−1)|R is trivial so Y containscomplete curves which do not intersect Z. We write d = (d1, d2) for d1[R]+d2[F ] ∈ Heff

2 (Y )and use q1, q2 for the corresponding quantum parameters.

The restriction of OZ(−2) to an (irreducible, complete) curve in Y of homology class(d1, d2) with d2 > 0 has degree −2d2 < 0 so its total space contains not complete curves


except the zero section. It follows that stable maps in homology class (d1, d2 > 0) haveimage contained in Z, so we have an equality

M0,n(Y, (d1, d2)) = M0,n(Y, (d1, d2)),

although the virtual classes are different. The restriction of TY to Z splits as TZ⊕OZ(−2)so from the construction of the virtual classes, we have

[M0,n(Y, (d1, d2))]vir = c2d2−1(R1π∗f∗OZ(−2)) a [M0,n(Z, (d1, d2))]vir

(when d2 > 0). Note that the expected dimension of M0,n(Y, (d1, d2)) is n + d1 whilethe expected dimension of M0,n(Z, (d1, d2)) is n − 1 + d1 + 2d2. Since the moduli spaceM0,n(Y, (d1, d2)) is already compact when d2 > 0, the formal virtual localization sum weused to define the GW invariants (summing over the “ladders” fixed by the torus action)is equal to

〈α〉d1,d2 =∫

[M0,n(Z,(d1,d2))]vir

e∗α ` c2d2−1(R1π∗f∗OZ(−2))

by the virtual localization formula [34]. In particular, these invariants vanish if the degreeof the insertions is less than the expected dimension.

It will be convenient to encode the “non-equivariant” invariants Wd1,d2 := 〈Y ⊗d13 〉d1,d2

(which do not depend on t1, t2) in the generating series

W (x, y) :=∑

d1,d2

Wd1,d2xd2yd1

d1!.

(The other non-equivariant invariants are determined from these by the Divisor Axiom.)The series W will be closely related to the generating series Z for the non-equivariantorbifold invariants of X .

Lemma 6.5.1. The invariants for the homology class (d, 0) (multiple covers of the rigidsection) vanish when d > 1 and are determined when d = 1 by the Divisor Axiom and thevalues

〈Y ⊗n3 〉1,0 =

(−1)n

2tn−11 (n ≥ 0).

Proof. The rigid section R can be contracted in Z (by Castelnuovo’s criterion, say), so anystable map in homology class (d, 0) has image contained in the total space of OZ(−2)|R,which is a trivial bundle with weight 2t1. Since proper curves are collapsed when mappingto an affine space, we simply have

M0,n(Y, (d, 0)) ∼= C2t1 ×M0,n(P1, d),

with the torus fixed part corresponding to the first coordinate being zero. The bundleR1π∗f∗OZ(−2) is certainly trivial on this space, so the contribution to the virtual normalbundle only comes from R1π∗f∗NR/Z . The formal localization sum for the invariant 〈α〉d,0

is therefore identified with1

2t1

∫

M0,n(P1,d)e∗α ` cd−1(R1π∗f∗NR/Z).

Now we just compute the invariant by the Divisor Axiom, noting that the evaluation mape takes values in R and Y2|R = −t1, Y3|R = −ht1. ¤


Remark 6.5.1. The above proof shows that the Gromov-Witten invariants of Y and Zfor the homology class (d, 0) are related by the formula

〈α〉d,0Z = 2t1〈α〉d,0

Y .

Lemma 6.5.2. W1,d = 〈Y3〉1,d = −1 for any d > 0.

Proof. This invariant is given by

〈Y3〉1,d =∫

M0,1(Z,(1,d))e∗1(gh)c2d−1(R1π∗f∗OZ(−2)),

which is an integral of the correct degree (2d + 1) over a compact space, therefore inde-pendent of linearizations. Recall the SES

0 → OZ(−2) → p∗1(OP1 ⊕ OP1(−1))⊗ OZ(−1) → p∗1O(−1) → 0

from the proof of Lemma 6.4.2 (the last term is more canonically L1⊗L2 = OP1(−1)⊗OP1).Let F := p∗1OP1(−1) ⊗ OZ(−1). Since the restriction of p∗1O(−1) to any curve of degree(1, d) has degree −1 we have

R0π∗f∗p∗1O(−1) = R1π∗f∗p∗1O(−1) = 0

soc2d−1(R1π∗f∗OZ(−2)) = cd(R1π∗f∗F )cd−1(R1π∗f∗OZ(−1)).

We will compute the integral

〈Y3〉1,d =∫

M0,1(Z,(1,d))e∗1(gh)cd(R

1π∗f∗F )cd−1(R1π∗f∗OZ(−1))

by localization using a new C∗ action which fixes only two points 0,∞ in P1 and actson their tangent spaces with weight 1,−1, respectively. This action fixes four pointsR0, R∞, S0, S∞ of Z, namely, the points R0, S0 lying over 0 ∈ P1 and in the sections Rand S := PO(−1), and the corresponding points R∞, S∞ lying over ∞ ∈ P1. We linearizeg, h, OZ(−1) and OP1(1) (these determine a linearization of F ) as in the table below.

P TPP1 Tp1 |P OZ(−1)|P p∗1OP1(1)|P F |P g|P h|PR0 1 1 0 1 −1 1 1S0 1 −1 1 1 0 0 1R∞ −1 2 0 0 0 1 0S∞ −1 −2 2 0 2 −1 0

This C∗ action has four invariant curves: the sections R,S, and the fibers F0, F∞ over0,∞.

For any fixed point(f, C, P ) ∈M0,1(Z, (1, d))C

∗,

every component of C is either collapsed by f or maps onto one of the invariant curveswith full ramification over its two fixed points. Since f∗[C] = (1, d), either (1) exactlyone component of C maps isomorphically to S or (2) exactly one component of C mapsisomorphically to R. In either situation we will also refer to the distinguished componentof C as R or S (respectively). We will also refer to the uniquely determined points of C inthese components and in f−1(R0), f−1(R∞), f−1(S0), f−1(S∞) simply as R0, R∞, S0, S∞.


Also let C0 := f−1(F0) and C∞ := f−1(F∞). Let d0 be the degree of f |C0 : C0 → F0 anddefine d∞ similarly. In situation (1), we have

C = C0

∐

S0

S∐

S∞

C∞

and d0 + d∞ = d− 1, while in situation (2) we have

C = C0

∐

R0

R∐

R∞

C∞

and d0 + d∞ = d. Since C has only one marked point, C0 is a point if and only if d0 = 0because of stability (similarly with 0 replaced by ∞).

In situation (1), if S0 is a node of C (equivalently: if d0 > 0), then resolving it yields anormalization sequence which can tensor with f∗F and take cohomology to get a SES

0 → F |S0 → H1(C, f∗F ) → H1(C0 ∪ S, f∗F )⊕H1(C∞, f∗F ) → 0.

Since c1(F |S0) = 0, no such locus contributes to the localization sum. In particular, insituation (1), any contibuting locus has f [C] ⊆ S ∪ F∞ ⊂ Z. But because of our choiceof linearizations for g, h, any contributing locus must have e1(P ) = R0, so no fixed locicontribute in situation (1).

In situation (2), if R∞ is a node of C (equivalently: if d∞ > 0), then resolving it yieldsa normalization sequence which can tensor with f∗F and take cohomology to get a SES

0 → F |R∞ → H1(C, f∗F ) → H1(C0 ∪R, f∗F )⊕H1(C∞, f∗F ) → 0.

Since c1(F |R∞) = 0, no such locus contributes to the localization sum. Now assume weare in situation (2) and d∞ = 0 (so that C∞ = R∞). Then resolving the node R0,tensoring with f∗F and taking cohomology gives a SES

0 → F |R0 → H1(C, f∗F ) → H1(C0, f∗F ) → 0

and resolving R0, tensoring with OZ(−1) and taking cohomology gives a SES

0 → OZ(−1)|R0 → H1(C, f∗OZ(−1)) → H1(C0, f∗OZ(−1)) → 0.

Next, notice that F0 has two C∗ fixed points (R0 and S0) and that c1(OZ(−1)|R0) = 0and c1(F |S0) = 0, so that, in the above situation the integrand vanishes if more than oneirreducible component of C0 has positive degree onto F0. We have now shown that, insituation (2), the only contributing loci satisfy the three conditions below.

(1) C∞ = R∞(2) C0 has exactly one non-collapsed component CNC

0∼= P1 mapping with degree d to

F0.(3) e1(P ) = R0

It is not hard to see that there is only one stable map C = (C, f, P ) (up to isomorphism)in the C∗ fixed locus satisfying these properties. The curve C has three irreducible com-ponents: R, CNC

0 , and another component CC0 containing the marked point P , contained

in C0, meeting CNC0 at a node N , and collapsed by f to R0.

The contribution to the localization sum from this fixed point can be worked out as onPage 291 of [23] to get the result. ¤Lemma 6.5.3. W2,d = 1

6d− 16d

3.


Proof. Apply the WDVV equation (Proposition 6.4.3) in degree (2, d) with (i, j, k, l) =(1, 1, 2, 3) to get

2t1t2〈Y1, Y1, Y0〉〈Y2, Y3, Y1〉+ 2t1t2〈Y1, Y1, Y1〉〈Y2, Y3, Y0〉−2〈Y1, Y1, Y2〉〈Y2, Y3, Y3〉 − 2〈Y1, Y1, Y3〉〈Y2, Y3, Y2〉−2t1〈Y1, Y1, Y1〉〈Y2, Y3, Y1〉

= 2t1t2〈Y1, Y2, Y0〉〈Y1, Y3, Y1〉+ 2t1t2〈Y1, Y2, Y1〉〈Y1, Y3, Y0〉−2〈Y1, Y2, Y2〉〈Y1, Y3, Y3〉 − 2〈Y1, Y2, Y3〉〈Y1, Y3, Y2〉−2t1〈Y1, Y2, Y1〉〈Y1, Y3, Y1〉

(notation for sums over splitting of degrees is suppressed). By Lemmas 6.4.1, 6.5.1, and6.5.2, many of these invariants vanish and we are left with

−2〈Y1, Y1, Y3〉1,0〈Y2, Y3, Y2〉1,d − 2〈Y1, Y1, Y3〉1,d1〈Y2, Y3, Y2〉1,d−d1

−2t1〈Y1, Y1, Y1〉1,0〈Y2, Y3, Y1〉1,d

= −2〈Y1, Y2, Y2〉0,0〈Y1, Y3, Y3〉2,d − 2〈Y1, Y2, Y3〉1,d1〈Y1, Y3, Y2〉1,d−d1

(summed over d1). We can evaluate these using the aforementioned lemmas to get

12d2 − 1

2d+

d∑

d1=1

(d− d1)2 = −W2,d +d∑

d1=1

d1(d− d1).

Solving for W2,d completes the proof. ¤

Theorem 6.5.4. The genus zero equivariant Gromov-Witten potential function for Y isdetermined by its evaluation at q1 = 0:

ΦY |q1=0 = −12y0y2y3 +

112t−11 t−2

2 y30 −

14y1y

22 +

14t−11 t−1

2 y20y1

− 112y32 −

∑

d≥1

d−3edy2qd2

+14(t1 + t2)y2

2y3 + (t1 + t2)∑

d≥1

d−2edy2y3qd2

together with the WDVV equations and the non-equivariant terms linear in q1:

ΦY |t1=t2=0 = ΦY |t1,t2=0,q1=0 − 12y3e

y1q1 −∑

d≥1

y3ey1edy2q1q

d2 .

Proof. This follows immediately from Lemmas 6.4.2, 6.5.1, and 6.5.2, together with theDivisor Axiom. ¤

Comparing the results of Theorem 6.3.3 and Theorem 6.5.4, we can prove

Theorem 6.5.5. The genus zero equivariant Gromov-Witten potential functions for theorbifold X = [O ⊕ O(−1)/Z2] and its crepant resolution Y are related (up to unstableterms) by the change of variables

ΦY (x0, x1, ix2, ix3, q,−1) = ΦX (x0, x1, x2, x3, q).

Proof. We will use the usual analytic continuation (c.f. [14]) below.


12

tan(x

2

)=

(i

2+

∞∑

d=1

iedixqd2

)∣∣∣∣∣q2=−1

(35)

Since the change of variables respects the modified inverse Poincare pairings gX , gY , itexchanges the WDVV equations satisfied by the two potential functions. It only remainsto check that the “initial conditions” displayed in Theorems 6.3.3 and 6.5.4 match up,which is a straightforward computation using Equation 35. ¤Remark 6.5.2. Just to be sure, one can check the equality asserted in 6.5.5 to higherorder by hand. For example, in ΦX , the non-equivariant terms of degree two in q aregiven by

116

sec4(x2

2

) x23

2e2x1q2

(Theorem 6.3.2). From Lemma 6.5.3, we have W2,d = 16d − 1

6d3, so in ΦY , the non-

equivariant terms of degree two in q1 are given by∑

d

(16d− 1

6d3

)x2

3

2e2x1edx2q21q

d2 .

One can check that these terms match up under the change of variables of Theorem 6.5.5by differentiating both sides of Equation 35 a couple of times and using the resultinganalytic continuations.

Conjecture 6.5.6. For any fixed n ≥ 1, the numbers Wn,d are given, for all d > 0, by apolynomial function of d.

I checked this for n ≤ 10.


7. Flags

After fixing a non-degenerate bilinear form on a vector space V we define a Z2 actionon the manifold of flags F in V by taking a flag to its orthogonal complement. When Vis of dimension 3 we prove the Crepant Resolution Conjecture 2.6.1 for the stack [F/Z2].We explicitly compute all degree zero and all 3-point invariants for the orbifold and theresolution, then argue that the other invariants are determined by WDVV and the DivisorAxiom. The quotient F/Z2 is contained in the quotient of P2 × P2 by the Z2-actioninterchanging the factors. The crepant resolution Y (a hypersurface in the Hilbert schemeHilb2P2) is the projectivization of a novel rank 2 vector bundle over P2.

The material in this section (Section 7) has mostly appeared already as [32]. Theonly major changes are the additional of Section 7.2 and an explanation of the change ofvariables (Section 7.11) in terms of Perroni’s change of variables (Theorem 5.0.4).

7.1. Setup. Fix a non-degenerate bilinear form 〈, 〉 on an n-dimensional complex vectorspace V . For a linear subspace A ⊆ V let

A⊥ := v ∈ V : 〈v, a〉 = 0 for all a ∈ Adenote the “orthogonal complement” of A with respect to 〈, 〉. This is of dimension n −dimA but may not be disjoint from A. We may identify Hom(V/A,C) with A⊥ via thisinner product. Then

(A1 ⊂ A2 ⊂ · · · ⊂ An−1) 7→ (A⊥n−1 ⊂ · · · ⊂ A⊥2 ⊂ A⊥1 )

is an involution of (Z2-action on) F , the manifold of complete flags in V . For example,consider the bilinear form

〈v, w〉 := vnw1 + vn−1w2 + · · ·+ v1wn

on Cn. Let W be the idempotent n × n matrix whose (i, j) entry is δn+1−ji , so that

multiplying by W on the right reverses the columns of a matrix and multiplying on the leftreverses the rows. The idempotent outer automorphism A 7→W (AT )−1W of G = SLn(C)preserves the Borel subgroup B+ of upper triangular matrices, hence induces a Z2-actionon F = G/B+ which takes a flag to its orthogonal complement with respect to the abovebilinear form.

The rest of this section is devoted to proving the CRC for X = [F/Z2] when V isof dimension 3. In particular, we will explicitly compute the small quantum cohomologyrings of X and the crepant resolution Y and give an explicit isomorphism between themin Section 7.11. We check that the genus zero potential functions agree in Section 7.12.

In Section 7.3 we recall some general facts about flag manifolds, then discuss the Z2-action on the manifold of flags in C3, determining the fixed locus and the correspondingrestriction map on cohomology, leading to a description of the Chen-Ruan orbifold co-homology ring of X . Then we find explicit descriptions of some simple moduli spacesof orbifold stable maps to X , which, together with associativity of the orbifold quan-tum product, we use to determine the 3-point Gromov-Witten invariants and the orb-ifold quantum cohomology ring of X . In Section 7.5 we describe the crepant resolutionr : Y → F/Z2 as a P1-bundle over P2 which is a hypersurface in the Hilbert schemeHilb2P2. In [55], J. Wise showed (using Graber’s computations [33]) that the CRC holdsfor X = [P2 × P2/Z2], Y = Hilb2P2. It may be possible to use this result together with


the Orbifold Quantum Lefschetz Hyperplane Theorem ([53], Section 5) to compute theorbifold potential of [F/Z2].

7.2. Some classical geometry. In order to provide a context for our geometric discussionit will be useful to have a diagram containing the varieties of interest. Continuing toassume V is of dimension 3, we write P2 for PV . Using our inner product on V , weview the complete flags F in V as the subvariety of P2 × P2 consisting of pairs (A,B)with 〈A,B〉 = 0. We could think of such an (A,B) as specifying the flag (A ⊂ B⊥) orthe flag (B ⊂ A⊥). Indeed, the “orthogonal complement” involution of F is induced bythe involution of P2 × P2 exchanging the factors. We have a commutative diagram withcartesian squares as below.

E

²²

Â Ä // Bl∆P2 × P2

²²

// Hilb2P2

²²

// BlP2P Sym2 V

²²

P(TP2|C)

²²

;;wwwwwwwÂ Ä // BlCF

²²

//+ ®

88rrrrrrrY

²²

::uuuuuuuu // BlCP4

²²

77pppppppp

P2 Â Ä // P2 × P2 // Sym2 P2 // PSym2 V

C

;;wwwwwwwwÂ Ä // F+ ®

88rrrrrrrrr // F/Z2

::uuuuuu// P4

77ppppppppp

Here C is the Z2 fixed locus in F . We will see in Section 7.3 that C ∼= P1 and thatthe composition of the arrows on the bottom row embeds C as a rational normal curvein P4. This P4 is the projectiviation of the (linear!) subspace of Sym2 V given by thekernel of 〈, 〉 : Sym2 V → C. It turns out that the quotient F/Z2 is the secant variety toC in P4. The space BlP2P Sym2 V is the blowup along the projectivization of image ofthe diagonal map V → Sym2 V taking v to v2 = [v ⊗ v]. The space E is P(N∆) = PTP2,which, coincidentally, is isomorphic to F . The space P(TP2|C) is just P1 × P1 since C isembedded as a conic in P2 to TP2|C splits as O(3) ⊕ O(3). The only space that remainsa bit mysterious is the crepant resolution Y !

In what follows, it turns out to be convenient to use a basis for the cohomology of Ythat does not so much reflect the geometry of this diagram. In order to reconcile thiswith the Perroni change of variables of Theorem 5.0.4, we will compute the cohomology ofevery variety in the diagram above and the maps between them. The uninterested readercan skip this.

Using p1, p2 to denote the two hyperplane classes in H2(P2 × P2), I claim that

H∗(Bl∆P2 × P2) = Z[p1, p2,H]/〈p31, p

32, p1H − p2H,H

2 + p21 + p2

2 + p1p2 − 3p1H〉where H is the Poincare dual of E. To see this, we first note that H|E = OE(−1) and

H∗(E) = Z[h,H]/〈h3,H2 − 3hH + 3h2〉by Leray-Hirsch, where h is the hyperplane class from P2 ∼= ∆. The defining property ofH is the integration rule ∫

Bl∆P2×P2

Hα =∫

Eα|E ,


from which we can determined the pairing

H4(Bl∆P2 × P2)⊗H4(Bl∆P2 × P2) → Z

α⊗ β 7→∫

Bl∆P2×P2

αβ.

Since it is given byp21 p2

2 p1p2 p1H p2H H2

p21 0 1 0 0 0 −1p22 1 0 0 0 0 −1

p1p2 0 0 1 0 0 −1p1H 0 0 0 −1 −1 −3p2H 0 0 0 −1 −1 −3H2 −1 −1 −1 −3 −3 −6

the two degree four relations are immediate and no other relations are needed because ofrank considerations.

We will see in the next section (7.3) that

H∗(F ) = Z[p1, p2]/〈p1p2 − p21 − p2

2, p21p2 − p1p

22〉

so we find by a similar calculation that

H∗(BlCF ) = Z[p1, p2,H]/〈p1p2− p21− p2

2, p21p2− p1p

22,Hp1−Hp2, H

2 + 2p21 + 2p2

2− 3Hp1〉with the restriction map

H∗(Bl∆P2 × P2) → H∗(BlCF )given in the obvious way.

Alternatively, since Bl∆P2 × P2 resolves the rational map

P2 × P2 99K P2∗(A,B) 7→ A+B

we could also view Bl∆P2 × P2 as PW ×P2∗ PW , where W is the tautological rank twobundle on P2∗. From this point of view, Leray-Hirsch and (relative) Kunneth give

H∗(Bl∆P2 × P2) = Z[p1, p2, T1]/〈T 31 , p

21 − T1p1 + T 2

1 , p22 − T1p2 + T 2

1 〉,where T1 is the hyperplane class from P2∗ and now we think of pi as being the pullback ofOPW (1) via the projection πi : PW×P2∗PW → PW . We have used that c(W ) = 1−T1+T 2

1 .(It is not difficult to see that the two descriptions of pi we have given are the same.)

With this description of Bl∆P2 × P2, it is clear that the Z2 action is given by ex-changing the factors in PW ×P2∗ PW , so the quotient Hilb2P2 is then PSym2W . LetT2 := c1(OPSym2 W (1)) ∈ H2(Hilb2P2). Then T2 pulls back via the Z2 quotient map top1 + p2; indeed, T2 and T1 generate the Z2-invariant subring of H∗(Bl∆P2 × P2).

I claim that the two presentions of H∗(Bl∆P2×P2) are related by the change of variables

H = p1 + p2 − T1.

We will prove this by brute force using localization, but there are probably better ways.View a point of Bl∆P2 × P2 as a triple (W,A,B) where W ∈ P2∗ and A,B ∈ PW . LetT = (C∗)3 act on V = C3 in the usual way. There is an induced T action on Bl∆P2 × P2

and the classes H,T1, p1, and p2 lift naturally to equivariant classes. The T action has


12 fixed points, and the restrictions of H,T1, p1, p2 to the equivariant cohomology of thefixed points are given in the table below.

P = (W,A,B) p1|P p2|P H|P T1|P(E01, E0, E1) −t0 −t1 0 t2(E01, E1, E0) −t1 −t0 0 t2(E01, E0, E0) −t0 −t0 −t0 t2(E01, E1, E1) −t1 −t1 −t1 t2(E02, E0, E2) −t0 −t2 0 t1(E02, E2, E0) −t2 −t0 0 t1(E02, E0, E0) −t0 −t0 −t0 t1(E02, E2, E2) −t2 −t2 −t2 t1(E12, E1, E2) −t1 −t2 0 t0(E12, E2, E1) −t2 −t1 0 t0(E12, E1, E1) −t1 −t1 −t1 t0(E12, E2, E2) −t2 −t2 −t2 t0

SinceH − p1 − p2 + T1 = t0 + t1 + t2

on every fixed point, it is pulled back from the equivariant cohomology of a point, and weget the desired equality on (nonequivariant!) cohomology classes.

Since H is the Poincare dual of E in Bl∆P2 × P2, it follows that

2H = 2(p1 + p2)− 2T1

is the Poincare dual of E in Hilb2P2.

7.3. Orbifold cohomology of X . Abusing notation, we let Ai denote the rank i vectorbundle on F whose fiber over (A1 ⊂ · · · ⊂ An−1) is Ai. Let ui := c1(Ai+1/Ai) and letpi := −c1(detAi). Applying adjunction to the short exact sequence (SES)

0 → Ai → Ai+1 → Ai+1/Ai → 0

we get −pi+1 = ui − pi (one should put a “dual” or a minus sign somewhere in the fifthsentence in the second paragraph of page 3 in [35]). Once a basis e1, . . . , en for V ischosen, the effective cone Heff

2 (F ) of F is spanned by the curve classes

Wi := (A1 ⊂ · · · ⊂ An−1) ∈ F : Aj = 〈e1, . . . , ej〉 unless j = i,which satisfy 〈Wi, pj〉 = δij so the pi are dual to the effective cone under the evaluationpairing. The Z2-action takes Wi to Wn−i (hence pi to pn−i in cohomology). Dualizing theSES

0 → Ai → V → V/Ai → 0we find that c1(Ai) = c1(A⊥i ).

Specialize to the case where V = C3. In terms of the pi, the cohomology ring of F canbe presented:

H∗(F,Z) = Z[p1, p2]/〈p1p2 − p21 − p2

2, p1p22 − p2

1p2〉(the relations are given by the elementary symmetric polynomials in the ui). The forgetfulmaps

π1(A1 ⊂ A2) := A1 ∈ P2

π2(A1 ⊂ A2) 7→ A2 ∈ P2∗ := Gr2(C3)


make F a P1-bundle over P2, P2∗ respectively. The classes p1, p2 are pullbacks of thepositive generators of H2(P2,Z), H2(P2∗,Z) via these projections.

The fixed locus C := FZ2 is given by

(A1 ⊂ A2) : A⊥2 = A1 = (A1 ⊂ A2) : A2 = A⊥1 = (A1 ⊂ A⊥1 ) : 〈A1, A1〉 = 0= (A⊥2 ⊂ A2) : 〈, 〉|A2 is degenerate.

This is a section of both of the above projections over a conic in P2,P2∗ so C ∼= P1 andthe restriction map H∗(F,Z) → H∗(C,Z) takes p1 and p2 to twice the positive generatorx of H2(C,Z) ∼= Z.

Next we compute the Poincare dual of [C] in H4(F,Z). This is characterized by theproperty that

〈a ∪ P.D.[C], [F ]〉 = 〈a|C , [C]〉 for all a ∈ H2(F,Z)

so, since we know 〈p21p2, [F ]〉 = 〈p1p

22, [F ]〉 = 1, we find that

P.D.[C] = 2p1p2.

From this we can compute the Chen-Ruan orbifold cohomology ring H∗orb(X ). Notice

that p1 + p2 generates H∗(F,Q)Z2 as a Q-algebra. Additively (with complex grading) wewill use the basis S0, . . . , S5 below:

grading/sector H∗(F,Q)Z2 H∗(C,Q)0 S0 = 11 S1 = p1 + p2 S2 = 12 S3 = (p1 + p2)2 = 3p1p2 S4 = x3 S5 = (p1 + p2)3 = 6p2

1p2

with multiplication S1S2 = (p1 + p2)|C = 4S4, S22 = P.D.[C] = (2/3)S3, S2S4 = (1/6)S5.

Thus the Poincare duality metric G = (Gij) = (〈SiSj , [F/Z2]〉) and the correspondingdual basis are

G =

0 0 0 0 0 30 0 0 3 0 00 0 0 0 1/2 00 3 0 0 0 00 0 1/2 0 0 03 0 0 0 0 0

S0 = (1/3)S5

S1 = (1/3)S3 S2 = 2S4

S3 = (1/3)S1 S4 = 2S2

S5 = (1/3)S0

and a nice presentation is H∗orb(X ) ∼= Q[S1, S2]/〈S3

2 , 3S22 − 2S2

1〉.

7.4. Orbifold quantum cohomology of X . We will identify an effective Z2-invarianthomology class aW1 + aW2 with the non-negative integer a. To compute the orbifoldquantum cohomology of X it will suffice to study the moduli spaces M0;0,3(X , 1) andM0;2,0(X , 1), which have expected dimension (this will be equal to the actual dimension)5 and 2 respectively. The quantum parameter q for QH∗

orb([F/Z2]) should have degree 2because

〈c1(TX ), 1〉 = (1/2)〈c1(F ),W1 +W2〉 = (1/2)〈2p1 + 2p2,W1 +W2〉 = 2.


We may identify the first of these moduli spaces (at least coarsely) with the usual Kont-sevich stable map space M0;3(F, (1, 0)) by the map

[f : (Σ, P1, P2, P3) → F ] 7→ [f∐f⊥ : Σ := Σ

∐Σ → F ]

in which case the evaluation maps ei : M0;0,3(X , 1) → F/Z2 are identified with evi +ev⊥i .Thus we have a commutative diagram

M0;3(F, (1, 0)) evi−−−−→ F

∼=y

yM0;0,3(X , 1) ei−−−−→ F/Z2

from which we can compute 〈S1, S3, S3〉1 = 9 and 〈S1, S1, S5〉1 = 6 (c.f. [35]).

Next we claim that the evaluation map

e = (e1, e2) : M0;2,0(X , 1) → C × C ∼= P1 × P1

is an isomorphism (coarsely). A point (f : Σ → F,Σ, R1, R2) ∈ M0;2,0(X , 1) parameter-izes a curve of minimal degree, so at most one component Σ1 of the base curve Σ is notcollapsed. Since there are only two marked points, either Σ = Σ1 or Σ = Σ1 qN Σ2 withR1, R2 ∈ Σ2 \ N (stability!).

In the first case, R1, R2 ∈ Σ ∼= P1 so Σ ∼= P1 and f : Σ → F is equivariant of bidegree(1, 1). In particular, f is an embedding so

e = ((A ⊂ A⊥), (B ⊂ B⊥)) ∈ C × C

with A 6= B. Now, notice that any equivariant map g : P1 → F is determined byπ1g : P1 → P2, so since π1f above is of degree 1, it is determined by A,B ∈ P2 hence thereis at most one map in M0;2,0(X , 1) with no collapsed components with e as above. Tosee that there is exactly one, let V := Span(A,B) so C3 = V ⊕ V ⊥ and consider the mapPV → F given by

C 7→ (C ⊂ C ⊕ V ⊥) ∈ F.This is an embedding with Z2-invariant image because

((C ⊕ V ⊥)⊥ ⊂ C⊥) = (D ⊂ D ⊕ V ⊥)

for some D ⊂ V . Notice that this curve is of bidegree (1, 1) and meets C exactly at thetwo coordinates of e.

When there is a collapsed component Σ2∼= P1, it contains the two marked points so

e = ((A ⊂ A⊥), (A ⊂ A⊥)) ∈ C × C

and the cover Σ is unramified over the uncollapsed component Σ1, so over Σ1, we havef |Σ1

: P1 q P1 → F with degrees (0, 1), (1, 0). Since f is equivariant, it is uniquelydetermined by, say, the degree (0, 1) map, which must be an isomorphism onto

(A ⊂ B) ⊂ F

(A fixed, B varying). The degree (1, 0) map must be an isomorphism onto

(C ⊂ A⊥) ⊂ F,

so the two preimages N1, N2 ∈ Σ of the node N ∈ Σ map to (A ⊂ A⊥), and Σ2 is a P1

glued to Σ1 at N1, N2 and collapsed by f to (A ⊂ A⊥). This proves the claim, so by the


Divisor Axiom (see Page 193 in [23] for the axioms of Gromov-Witten theory used here)we compute

〈S1, S4, S4〉1 = (1/2)〈p1 + p2,W1 +W2〉∫

C×Cπ∗1x · π∗2x = 1.

For dimension reasons, only nine 3-point Gromov-Witten numbers could possibly benon-zero. They turn out to be:

Nontwisted (r = 0) Twisted (r = 2)〈S1, S3, S3〉1 = 9 〈S1, S4, S4〉1 = 1〈S1, S1, S5〉1 = 6 〈S2, S3, S4〉1 = 3〈S3, S3, S5〉2 = 54 〈S2, S2, S5〉1 = 6〈S1, S5, S5〉2 = 36 〈S4, S4, S5〉2 = 3〈S5, S5, S5〉3 = 0

We computed 3 of these above, and the rest can be derived from associativity of the smallquantum orbifold cohomology ring, by considering the following associativity checks inorder:

(S1 ? S1) ? S2 = S1 ? (S1 ? S2) =⇒ 〈S2, S3, S4〉1 = 3(S2 ? S3) ? S1 = S2 ? (S3 ? S1) =⇒ 〈S2, S2, S5〉1 = 6(S3 ? S2) ? S2 = S3 ? (S2 ? S2) =⇒ 〈S3, S3, S5〉2 = 54(S1 ? S3) ? S5 = S1 ? (S3 ? S5) =⇒ 〈S1, S5, S5〉2 = 36

=⇒ 〈S5, S5, S5〉3 = 0(S4 ? S1) ? S3 = S4 ? (S1 ? S3) =⇒ 〈S4, S4, S5〉2 = 3

The same result can be obtained from the orbifold WDVV equation6. These 3-pointinvariants form, in the usual way, a commutative (associative!) graded ring structure onH∗

orb(X )⊗Q Q[q] whose multiplication table is given below.

S1 ? S1 = S3 + 2q S2 ? S2 = (2/3)S3 + 2qS1 ? S2 = 4S4 S2 ? S3 = 6qS2

S1 ? S3 = S5 + 3qS1 S2 ? S4 = (1/6)S5 + qS1

S1 ? S4 = 2qS2 S2 ? S5 = 12qS4

S1 ? S5 = 2qS3 + 12q2

S3 ? S3 = 3qS3 + 18q2 S4 ? S4 = (1/3)qS3 + q2

S3 ? S4 = 6qS4 S4 ? S5 = 6q2S2

S3 ? S5 = 18q2S1 S5 ? S5 = 12q2S3

Deforming the relations in H∗orb(X ) we can give a presentation:

QH∗orb(X ) = Q[S1, S2, q]/〈S3

2 − 6qS2, 3S22 − 2S2

1 − 2q〉

7.5. The crepant resolution Y . After fixing a non-degenerate symmetric bilinear form〈, 〉 : C3⊗C3 → C we may regard the smooth variety F of complete flags in C3 as the sub-space of P2 × P2 consisting of pairs (A,B) with 〈A,B〉 = 0. The Z2-action interchangingthe factors of P2 × P2 restricts to the Z2-action on F taking a flag to its orthogonal com-plement. The rational map P2×P2 99K P2∗ taking two (distinct) 1-dimensional subspacesof C3 to the 2-dimensional subspace they span is undefined on the diagonal, but can beresolved by blowing up the diagonal (the Z2-action lifts to an action on the blowup). The

6It is also interesting to compute the degree 1 twisted invariants by localization.


fiber of the resulting map p : Bl∆(P2 × P2) → P2∗ over W ∈ P2∗ is canonically PW × PW ,while the fiber p−1(W ) ∩ F of the restriction of p to the proper transform F of F is

(A,B) ∈ PW × PW : 〈A,B〉 = 0.The topology of this fiber depends on whether W⊥ ⊂W . If so, then the fiber is just

(W⊥, A) : A ⊂W ∪(W⊥,W⊥) (A,W⊥) : A ⊂W ∼= P1 ∨ P1,

while if W⊥ ∩W = (0), then the fiber is the graph of the idempotent automorphism ofPW taking A to A⊥ ∩W . The map p is Z2-equivariant, so it descends to a map on theZ2-quotients. The fiber of π over W is of course Sym2 PW ∼= P Sym2W . T. Graber [33]used this to note that Hilb2P2 = Bl∆(P2×P2)/Z2 is the projectivization of Sym2W , whereW now denotes the tautological rank 2 bundle on P2∗. Since the first Chern class of O3/Wis the positive generator T1 of H2(P2∗,Z) ∼= Z, the SES

0 →W → O3 → O3/W → 0

shows c(W ) = 1− T1 + T 21 from which one easily computes c(Sym2W ) = 1− 3T1 + 6T 2

1 .We will argue in a moment that Y := F /Z2 is a crepant resolution of singularities ofF/Z2. For now, notice that the fiber of π|Y over W is canonically the projectivization ofthe 2-dimensional space

V := w1 ∨ w2 ∈ Sym2W : 〈w1, w2〉 = 0 ⊂ Sym2W

so that we may describe Y as the projectivization of the corresponding rank 2 vectorbundle V ⊂ Sym2W on P2∗. The quotient bundle Sym2W/V is isomorphic to the trivialbundle by the map [w1 ∨ w2] 7→ 〈w1, w2〉 so we have c(V ) = c(Sym2W ) = 1− 3T1 + 6T 2

1 .

The proper transform F of F ⊂ P2 × P2 in Bl∆(P2 × P2) is obtained by blowing up Falong K = (F ∩∆) ∼= P1, which is the locus of flags of the form (W⊥ ⊂W ) (i.e. the fixedlocus of the Z2-action on F ).

Applying the Leray-Hirsch theorem to the projective bundle description of Y yields apresentation of its cohomology ring

H∗(Y,Z) = Z[T1, T2]/〈T 31 , T

22 − 3T1T2 + 6T 2

1 〉where T1 is pulled back from P2∗ and T2 = c1(OPV (1)). We can also compute the Chernclasses c(TY ) = 1 + 2T2 − 6T 2

1 + 6T1T2 + 6T 21 T2.

7.6. Curve classes in Y . Restricting PV to a line in P2∗ yields a Hirzebruch surfacewhose algebraic type turns out to depend on the type of line as follows. A generic line isof the form

À := B ∈ P2∗ : A ⊂ Bfor some fixed 1-dimensional A ⊂ C3 where A ∩ A⊥ = (0). Consider the line bundleL ⊂ W |À

over À whose fiber over B ∈ À is B ∩ A⊥. Choose some nonzero a ∈ A andnotice that 1 7→ a gives an injection O → W |À

, yielding a splitting W |À∼= O ⊕ L, so

c1(L) = −1. There is also an injective map of vector bundles on ÀL→ V |À

b 7→ a ∨ band V |À

has first Chern class −3 so the quotient line bundle has first Chern class −2,hence the corresponding SES is split for cohomological reasons and we get

V |À∼= O(−1)⊕ O(−2).


We conclude that the projectivization of V |Àis the Hirzebruch surface F1.

Now consider some A ∈ P2 with A ⊂ A⊥ (equivalently 〈A,A〉 = 0). We call thecorresponding line À ⊂ P2∗ a jump line. Here we may simply take a nonzero a ∈ A andget an injective vector bundle morphism O → V |À

by taking 1 → a∨a. The quotient hasfirst Chern class −3 so the SES is again split for cohomological reasons and we have

V |À∼= O ⊕O(−3)

so PV |Àis the Hirzebruch surface F3. On jump lines, the dimensions of H0(À, V |À

) andH1(À, V |À

) jump up from 0, 1 (for a generic line) to 1, 2 respectively.

We now study rational curves in Y . Identify a curve class [C] with the pair

(〈T1, [C]〉, 〈T2, [C]〉) ∈ Z2.

Using WDVV (see [23] or [5] for orbifolds) we will see that the 3-point Gromov-Wittennumbers of Y can be computed by studying only the moduli spaces M0;2(Y, (0, 1)),M0;3(Y, (0, 2)), and M0;0(Y, (n, 0)). Curves corresponding to pairs (0, a) are collapsedby π and are thus (branched) covers of some fiber F of π. Since 〈T2, [F ]〉 = 1 we alwayshave a ≥ 0 and the moduli space

M0;n(Y, (0, a))

is a fiber bundle over P2∗ whose fiber over W is M0;n(PW,a). This is smooth (as a stack)of the expected dimension

dimY − 3 + n+ 〈c1(TY ), (0, a)〉 = n− 2a.

In particular,M0;2(PW, 1) = PW × PW

so we have a simple description of M0;2(Y, (0, 1)) and its evaluation maps, from which wecan easily evaluate the 2-point Gromov-Witten invariants for the homology class (0, 1).The only such invariants which are non-zero are

〈T2, T21 T2〉0,1 = 1 and 〈T1T2, T1T2〉0,1 = 1.

Using the Divisor Axiom, we can also evaluate all 3-point invariants for this homologyclass.

Since M0;3(Y, (0, 2)) is a fiber bundle over P2∗ we know that the 3-point invariants ofthe form 〈T i

1Tl2, T

j1T

m2 , T

k1 T

n2 〉0,2 will vanish if i + j + k > 2. Combining this with the

Dimension Axiom shows that all such 3-point invariants are zero.

Now we turn our attention to curves contained in one of the Hirzebruch surfaces men-tioned above. The effective cone of a Hirzebruch surface is generated by the fiber class andthe class of the rigid section: any Hirzebruch surface can be written as P(O⊕O(n)) wheren ≤ 0, and the rigid section s is obtained by taking the subspace spanned by the trivialfactor. As long as n 6= 0 this is the unique section in its homology class. The rigid sectionhas normal bundle O(n). One can easily check that the rational curve corresponding tothe rigid section s of the Hirzebruch surface F1 over a generic line À corresponds to thehomology class (1, 1), while the rational curve corresponding to the rigid section t of F3

over a jump line À is in the homology class (1, 0).

Now consider the moduli spaceM0;0(Y, (1, 0)). The rigid section t of F3 is unique, so thismoduli space is the space of jump lines, which is a P1 given by the conic S2 ⊂ P2 consisting


of those A ∈ P2 with 〈A,A〉 = 0. We may explicitly identify M1 := M0;1(Y, (1, 0)) as well.The map π1 forgetting the marked point makes M1 a P1-bundle over M0;0(Y, (1, 0)) ∼=S2

∼= P1. I claim this is a trivial Hirzebruch surface. First of all, for A ∈ S2, giving apoint of the section tA is the same as giving a point of the line `A, which is the same asgiving a 1-dimensional subspace of C3/A. However, it is a simple matter to check thatthere is a canonical isomorphism P(C3/A) = PHom(A,C3/A), so we need only show thatthe restriction of the tangent bundle of P2 to S2 has balanced splitting type. Indeed, thisis a special case of the well-known fact that the restriction of TPn to any rational normalcurve (image of a degree n embedding P1 → Pn) has balanced splitting type.

The expected dimension of the moduli space M0;0(Y, (1, 0)) is 0 (this is a general phe-nomenon for crepant resolutions—see below), so we must identify the virtual fundamentalclass. This is given by the (Poincare dual of the) first Chern class of the vector bundle overS2 whose fiber over A is H1(tA, NtA/Y ). We will soon check that the rank of this vectorbundle is 1 (the excess dimension). The moduli space M1 is contained in Y as the unionof all rigid sections over jump lines (these are disjoint). Since M1 is a trivial Hirzebruchsurface, the normal bundle of a rigid section tA in M1 is trivial. The SES

0 → TtA → TY |tA → NtA/Y → 0

implies that c1(NtA/Y ) = −2. Thus the SES

0 → NtA/M1∼= OtA → NtA/Y → NM1/Y |tA → 0

must be split so that NtA/Y∼= OtA ⊕ OtA(−2) and we have a canonical isomorphism

H1(tA, NtA/Y ) = H1(tA, NM1/Y |tA).

We will see in Section 7.7 that M1 ⊂ Y is the exceptional divisor of the resolution mapr : Y → F/Z2, and that the rigid sections tA are the exceptional fibers of r. In particular,it follows from Theorem 3.0.2 that NM1/Y |tA = OtA(−2) so h1(tA, NM1/Y |tA) = 1. We willnow give an alternative proof of this.

We introduce an algebraic C∗-action on Y by taking a maximal torus in the Lie groupSO3(V, 〈, 〉) of matrices preserving 〈, 〉, which acts naturally on Y . Explicitly, we may takethe C∗-action

λ · (z0, z1, z2) := (λz0, z1, λ−1z2)

on C3, which preserves the bilinear form

〈(z0, z1, z2), (w0, w1, w2)〉 := z0w2 + z1w1 + z2w0

mentioned in the introduction. The jump lines fixed by this action correspond to thepoints [1 : 0 : 0] ∈ S2 and [0 : 0 : 1] ∈ S2. This C∗-action on Y has 6 fixed points. Thecohomology classes T1, T2, and c1(NM1/Y ) naturally lift to equivariant cohomology classes(M1 is invariant under the action so the last class lies in H∗

C∗(M1)) whose weights at thefixed points are listed in the table below.


P ∈ Y C∗ TPY T1|P T2|P NM1/Y |PP1 := 〈e0 ∨ e0〉 ⊂W |E01 ⊂ Sym2E01 −1,−2,−1 −1 −2 −2P2 := 〈e0 ∨ e1〉 ⊂W |E01 ⊂ Sym2E01 −1,−2, 1 −1 −1P3 := 〈e2 ∨ e2〉 ⊂W |E02 ⊂ Sym2E02 1,−1, 4 0 2 4P4 := 〈e0 ∨ e0〉 ⊂W |E02 ⊂ Sym2E02 1,−1,−4 0 −2 −4P5 := 〈e1 ∨ e2〉 ⊂W |E12 ⊂ Sym2E12 1, 2,−3 1 1P6 := 〈e2 ∨ e2〉 ⊂W |E12 ⊂ Sym2E12 1, 2, 1 1 2 2

We will use this to show that NM1⊂Y∼= OP1×P1(6,−2). The map

[s : t] 7→ 〈e0 ∨ e0〉 ⊂W |Span(e0,(0,s,t)) ∈M1

is the inclusion of the fiber of π1 over E0 ∈ S2 ⊂ P2. This fiber contains two fixed points:P1 and P4. From the localization chart we can see that

〈c1(NM1/Y ), [π−11 (E0)]〉 = −2

so NM1/Y = OP1×P1(d,−2) for some integer d. The map

[s : t] 7→ 〈(s2,√−2st, t2) ∨ (s2,√−2st, t2)〉 ⊂W |Ker(z 7→〈z,(s2,

√−2st,t2)〉) ∈M1

is a C∗-invariant section of π1 containing the fixed points P1 and P3, so it is either aC∗-invariant curve in M1 of degree (1, 1) or it is a fiber of π2. The integral of c1(NM1/Y )over this curve is 6, so in the first case we would have NM1⊂Y

∼= OP1×P1(8,−2) and in thesecond case we would have NM1⊂Y

∼= OP1×P1(6,−2). However, the first case is impossiblebecause then the fiber of π2 containing P1 would also contain P6 and thus the integralover this fiber would be 4 (not 8), whereas, in the second case, the curve containing P1

and P6 is of degree (1, 1) and the integral of c1(NM1/Y ) is correctly given by 4.

7.7. The resolution map r : Y → F/Z2. We may explicitly describe the crepant reso-lution map r : Y → F/Z2 as follows. A point of Y is specified by some W ∈ P2∗ togetherwith a subspace A = 〈a ∨ b〉 ⊂ V |W . The image of this point under r will be

[〈a〉 ⊂ 〈b〉⊥] = [〈b〉 ⊂ 〈a〉⊥] ∈ F/Z2.

It is easy to check that this is well-defined. To see that this is an isomorphism on the locusof [A1 = 〈a〉 ⊂ A2] ∈ F/Z2 where A2 6= A⊥1 , just notice that A⊥1 ∩ A2 is a 1-dimensionalsubspace of C3, spanned, say, by b so that

r−1([A1 = 〈a〉 ⊂ A2]) = 〈a ∨ b〉 ∈ V |〈a,b〉.

Recall that we constructed the rigid section over a jump line `A by always taking the1-dimensional subspace [a∨ a] in the fiber and letting only the 2-dimensional subspace Wvary (over all 2-dimensional subspaces containing A). Thus the resolution map collapsesthe rigid section over a jump line to a point so that the corresponding homology class (1, 0)is on the boundary of the effective cone of Y . Since the projection to P2∗ collapses the fiberclass, it also lies on the boundary of the effective cone, thus Heff

2 (Y,Z) = (a, b) : a, b ≥ 0.Since r is a crepant resolution (i.e. ωY = r∗ωF/Z2

) we expect to have a degree 0 quantumparameter because the canonical bundle of Y will evaluate 0 on the curve class collapsedby r (here: the class (1, 0)). The same phenomenon occurs for the crepant resolutionHilb2P2 → Sym2 P2 where the class of 0-dimensional subschemes supported at a fixedpoint is collapsed.


7.8. Classical geometry of the quotient. Here we make contact with some classicalgeometry by giving another description of the quotient F/Z2 and the resolution r : Y →F/Z2; none of this is strictly necessary in what follows. Recall that F is Z2-equivariantlyembedded in PV × PV (with Z2 exchanging the factors) as the set (A,B) : 〈A,B〉 = 0.Then

(PV × PV )/Z2 → PSym2 V

via the Segre embedding. The image of F/Z2 under this embedding inside the image of(PV × PV )/Z2 is given by a single linear equation. For example, if V = C3 with the“back-to-front” inner product from the introduction, then the Segre embedding is

[a : b : c] + [d : e : f ] 7→ [ad : ae+ bd : be : af + cd : cf : bf + ce]

and the image of F/Z2 is the intersection of the image of (P2×P2)/Z2 with the hyperplaneH = V (X2 + X3) ∼= P4. The fixed locus C ⊂ F is a curve in P2 × P2 of bidegree (2, 2)so its image (or rather, the image of C/Z2 = C ⊂ F/Z2) under the Segre embedding is arational normal curve in H. In the above coordinates,

C = [2s2 : 2ist : t2] + [2s2 : 2ist : t2]and the rational normal curve is

[4s4 : 8is3t : −4s2t2 : 4s2t2 : t4 : 4ist3].We will show that the image of F/Z2 is the secant (or chordal) variety of this rationalnormal curve in H, which is a singular degree 3 hypersurface (see page 120 in [38]). Indeed,taking the SO3-action into account, it is enough to show that the image of F/Z2 containsthe line between, say, the images of [1 : 0 : 0] + [1 : 0 : 0] and [0 : 0 : 1] + [0 : 0 : 1], as wellas the tangent line to the rational normal curve at, say, the image of [1 : 0 : 0] + [1 : 0 : 0].This first line is [s : 0 : 0 : 0 : t : 0], which is the image of

[√s : 0 :√−t] + [

√s : 0 : −√−t] ⊂ F/Z2.

The second line is [s : t : 0 : 0 : 0 : 0], which is the image of [1 : 0 : 0] + [s : t : 0].Every line in the chordal variety F/Z2 is either (1) a tangent line to the rational normal

curve or (2) a line connecting two distinct points on the rational normal curve. Takingthe SO3-action into account and considering the explicit computations above we see thatthe preimage of a line of type (1) in F consists of two smooth rational curves of degree(1, 0) and (0, 1) exchanged by the Z2-action and meeting at a point of C; the preimage ofa line of type (2) in F is a Z2-invariant smooth rational curve of degree (1, 1) meeting Cat two distinct points.

7.9. Gromov-Witten theory of Y . In this section, we compute the 3-point Gromov-Witten invariants of Y via WDVV. Using the basis Ti and its Poincare dual basis T ibelow

T0 = 1T1 T2

T3 = T 21 T4 = T1T2

T5 = T 21 T2

T 0 = T5

T 1 = T4 − 3T3 T 2 = T3

T 3 = T2 − 3T1 T 4 = T1

T 5 = T0


for the cohomology of Y , the Poincare duality metricG = (Gij) and its inverseG−1 = (Gij)are given as below.

G =

0 0 0 0 0 10 0 0 0 1 00 0 0 1 3 00 0 1 0 0 00 1 3 0 0 01 0 0 0 0 0

G−1 =

0 0 0 0 0 10 0 0 −3 1 00 0 0 1 0 00 −3 1 0 0 00 1 0 0 0 01 0 0 0 0 0

WDVV says that for any φ1, φ2, φ3, φ4 ∈ H∗(Y ), any n ≥ 0, any γ1, . . . , γn ∈ H∗(Y ), andany 2-dimensional homology class β we have

∑

a,bβ1+β2=βV ∪W=[n]

〈φ1, φ2, Ta, γv1 , . . . , γvk〉β1Gab〈φ3, φ4, Tb, γw1 , . . . , γwn−k

〉β2

=∑

a,bβ1+β2=βV ∪W=[n]

〈φ1, φ4, Ta, γv1 , . . . , γvk〉β1Gab〈φ2, φ3, Tb, γw1 , . . . , γwn−k

〉β2 .(36)

The terms on the LHS where either β1 or β2 is zero sum to give

LHSβ := 〈φ1, φ2, φ3φ4, γ1, . . . , γn〉β + 〈φ3, φ4, φ1φ2, γ1, . . . , γn〉β(37)

and similarly the terms on the RHS with β1 or β2 equal to zero give

RHSβ := 〈φ1, φ4, φ2φ3, γ1, . . . , γn〉β + 〈φ2, φ3, φ1φ4, γ1, . . . , γn〉β.(38)

In the remainder of this section we will always use this equation with n = 0 and with theinsertions φ1, φ2, φ3, φ4 equal to some Ti, Tj , Tk, Tl, so we will specify a WDVV equationby indicating the choice of β and (i, j, k, l).

We begin by computing all Gromov-Witten invariants for homology classes of the form(n, 0). By the Divisor and Dimension Axioms we need only compute the 0-point invariant〈〉n,0. Since (1, 0) is collapsed by the resolution map r, the curves of class (n, 0) are justn-fold branched covers of rigid sections over jump lines. Such a map is specified by a pointin the first factor of

M1 = M0;0(Y, (1, 0)) ∼= P1 × P1

together with an element of M0;0(P1, n) so that the moduli space M0;0(Y, (n, 0)) (whichhas expected dimension 0) is a product P1 ×M0;0(P1, n). Furthermore, we can identifythe obstruction class as the Euler class of the vector bundle

π∗1OP1(6)⊗ π∗2H1(C, f∗OP1(−2))

where, by abuse of notation, the second factor denotes the vector bundle on M0;0(P1, n)whose fiber over (f, C) is H1(C, f∗OP1(−2)). Recall that for a 2-dimensional vector spaceV we have a natural SES

0 → OPV (−1) → V → V/OPV (−1) → 0

which we can twist by OPV (−1) and pull-back by any map f to get a SES

0 → f∗OPV (−2) → V ⊗ f∗OPV (−1) → f∗[(V/OPV (−1))⊗OPV (−1)] → 0


where the vector bundle on the right is (non-canonically) trivial. The associated LES incohomology gives

0 → H0(C,OC) → H1(C, f∗OPV (−2)) → H1(C, V ⊗ f∗OPV (−1)) → 0.

Thus we compute

〈〉n,0 =∫M0;0(Y,(n,0)) e(Obstruction Bundle)

=∫P1×M0;0(P1,n) c2n−1(π∗1OP1(6)⊗ π∗2H

1(C, f∗OP1(−2)))= 6

∫M0;0(P1,n) c2n−2(H1(C, f∗O(−1)⊕ O(−1)))

= 6/n3

where the last equality is the Aspinwall-Morrison formula ([23], Theorem 9.2.3). Thus theonly non-zero 3-point invariant for the homology class (n, 0) is

〈T1, T1, T1〉n,0 = 6.

Next we compute the 3-point invariants when β = (n, 1). Using the Dimension andDivisor Axioms we see that it suffices to compute the 4 numbers

an225 := 〈T2, T2, T5〉n,1 an

233 := 〈T2, T3, T3〉n,1

an234 := 〈T2, T3, T4〉n,1 an

244 := 〈T2, T4, T4〉n,1

because the only other such 3-point invariants that might be non-zero are then determinedby the Divisor Axiom. For example

〈T1, T1, T5〉n,1 = n2〈T5〉n,1 = n2〈T2, T2, T5〉n,1 = n2an225

and similarly:〈T1, T2, T5〉n,1 = nan

225 〈T1, T3, T3〉n,1 = nan233

〈T1, T3, T4〉n,1 = nan234 〈T1, T4, T4〉n,1 = nan

244

We computed these invariants for n = 0 in Section 7.5. We can get a system of fourequations yielding a recursive formula (in n) for these 4 invariants by applying WDVVwith β = (n, 1) and (i, j, k, l) = (1, 1, 2, 3), (1, 1, 2, 4), (1, 2, 2, 3), (1, 2, 2, 4). Writing eachof these equations in the form

LHSβ − RHSβ = C

where C is determined by 3-point invariants for homology classes α < (n, 1) (in at leastone coordinate) we get a system of 4 equations

n2an225 +an

233 −nan234 = Cn

1

(3n2 − n)an225 +an

234 −nan244 = Cn

2

nan225 +6nan

233 +(1− 3n)an234 = 0

(3n− 1)an225 +6nan

234 +(1− 3n)an244 = 0

which uniquely determines the 4 unknowns in terms of Cn1 and Cn

2 when n > 0 becausethe determinant of the coefficient matrix is n2(6n− 1)(3n2 − 6n+ 1). The coefficients Cn

1

and Cn2 are easily worked out because if β1 + β2 = (n, 1), then one of β1 or β2 is of the

form (d, 0) and the corresponding three point invariant is almost always 0. We get

Cn1 = 6

∑n−1d=0 (3ad

233 − ad234) Cn

2 = 6∑n−1

d=0 (3ad234 − ad

244) .

Persevering a little, we work things out by hand for n = 1, 2 to find

a0225 = 1 a1

225 = 4 a2225 = 1

a0233 = 0 a1

233 = 1 a2233 = 4

a0234 = 0 a1

234 = 5 a2234 = 10

a0244 = 1 a1

244 = 19 a2244 = 25


C11 = 0 C2

1 = −12 C31 = 0

C12 = −6 C2

2 = −30 C32 = 0

so it follows that the invariants a3225, a

3234, a

3233, a

3244 vanish because the RHS of the above

system of equations is zero and the coefficient matrix is invertible. Applying the sameargument inductively (using Cn

1 = Cn−11 +18an−1

233 −6an−1234 and Cn

2 = Cn−12 +18an−1

234 −6an−1244 )

shows that all 3-point invariants for the class (n, 1) vanish when n > 2.

Next we compute the 3-point invariants for β = (n, 2). By the Dimension and DivisorAxioms it suffices to determine the 4 numbers

bn255 := 〈T2, T5, T5〉n,2 bn335 := 〈T3, T3, T5〉n,2

bn345 := 〈T3, T4, T5〉n,2 bn445 := 〈T4, T4, T5〉n,2

because 〈T1, T5, T5〉n,2 = (n/2)bn255. Apply WDVV with β = (n, 2) and (i, j, k, l) =(1, 4, 3, 3), (2, 4, 3, 3), (2, 3, 4, 4), (1, 4, 5, 2). In each of these cases, if β1 + β2 = (n, 2)and β1, β2 6= 0 then each summand in the WDVV equation will vanish unless β1 = (d, 1)and β2 = (n − d, 1) for some d ∈ 0, . . . , n. This is because the 3-point invariants for ahomology class (d, 0) (d > 0) vanish except 〈T1, T1, T1〉d,0, but this never appears in theabove WDVV equations because at most one of i, j, k, l is 1, so at most two insertions inany invariant are T1. Simplifying a little bit we find

bn335 =∑n

d=0(2d− n)ad233a

n−d234 + d(n− d)(ad

234an−d234 − ad

244an−d233 )

bn345 =∑n

d=0(n− d)(3d− 1)(ad234a

n−d234 − ad

244an−d233 )

bn445 =∑n

d=0(3n− 3d− 1)(3d− 1)(ad234a

n−d234 − ad

244an−d233 ) + (2d− n)ad

234an−d244

bn255 = bn445 +∑n

d=0 d(3n− 3d− 1)ad234a

n−d225 − d(n− d)ad

244an−d225

which implies that all these invariants vanish for n > 4 (because of the vanishing of thean’s for n > 2). Working the rest out by hand, we find that they also vanish when n = 4;the others are given below:

b0335 = 0 b1335 = 0 b2335 = 6 b3335 = 8b0345 = 0 b1345 = 1 b2345 = 20 b3345 = 21b0445 = 0 b1445 = 7 b2445 = 64 b3445 = 55b0255 = 0 b1255 = 2 b2255 = 8 b3255 = 2

Previously, we gave a geometric reason for the vanishing of the (0, 2) invariants.

Finally we compute the 3-point invariants 〈T5, T5, T5〉n,3. For dimension reasons, thisis the only 3-point invariant for β = (n, 3); when β = (n, k) with k > 3 then all 3-pointinvariants vanish for dimension reasons. To compute these invariants just apply WDVVwith β = (n, 3) and (i, j, k, l) = (2, 3, 5, 5) to get:

〈T5, T5, T5〉n,3 =n∑

d=0

3〈T2, T3, T3〉d,1〈T1, T5, T5〉n−d,2 − 〈T2, T3, T4〉d,1〈T1, T5, T5〉n−d,2

−〈T2, T3, T3〉d,1〈T2, T5, T5〉n−d,2 − 3〈T1, T2, T5〉d,1〈T3, T3, T5〉n−d,2

+〈T2, T2, T5〉d,1〈T3, T3, T5〉n−d,2 + 〈T1, T2, T5〉d,1〈T3, T4, T5〉n−d,2.

Surely the RHS vanishes if n > 5. In fact, an explicit calculation shows that theseinvariants vanish when n = 0, 1, 5 as well. The rest of the cases can be easily computedfrom our previous results (see below for the values).


7.10. Quantum cohomology of Y . Here we assemble the 3-point invariants computedin the previous section to give a multiplication table and a presentation of the (small)quantum cohomology ring of Y . This is a ring structure on H∗(Y,Q) ⊗ Q[q2][[q1]] withmultiplication given by

Ta ? Tb = TaTb +∑

(n,i)6=(0,0)

5∑

c=1

〈Ta, Tb, Tc〉n,iT cqn1 q

i2

(this is a finite sum for (a, b) 6= (1, 1)). The non-zero 3-point invariants of Y for non-zerohomology classes are given, up to reordering, by 〈T1, T1, T1〉n,0 = 6 for n > 0 and

〈T2, T2, T5〉0,1 = 1 〈T2, T4, T4〉0,1 = 1 〈T1, T1, T5〉1,1 = 4 〈T1, T2, T5〉1,1 = 4〈T1, T3, T3〉1,1 = 1 〈T1, T3, T4〉1,1 = 5 〈T1, T4, T4〉1,1 = 19 〈T2, T2, T5〉1,1 = 4〈T2, T3, T3〉1,1 = 1 〈T2, T3, T4〉1,1 = 5 〈T2, T4, T4〉1,1 = 19 〈T1, T1, T5〉2,1 = 4〈T1, T2, T5〉2,1 = 2 〈T1, T3, T3〉2,1 = 8 〈T1, T3, T4〉2,1 = 20 〈T1, T4, T4〉2,1 = 50〈T2, T2, T5〉2,1 = 1 〈T2, T3, T3〉2,1 = 4 〈T2, T3, T4〉2,1 = 10 〈T2, T4, T4〉2,1 = 25

〈T1, T5, T5〉1,2 = 1 〈T1, T5, T5〉2,2 = 8 〈T1, T5, T5〉3,2 = 3〈T2, T5, T5〉1,2 = 2 〈T2, T5, T5〉2,2 = 8 〈T2, T5, T5〉3,2 = 2

〈T3, T3, T5〉2,2 = 6 〈T3, T3, T5〉3,2 = 8〈T3, T4, T5〉1,2 = 1 〈T3, T4, T5〉2,2 = 20 〈T3, T4, T5〉3,2 = 21〈T4, T4, T5〉1,2 = 7 〈T4, T4, T5〉2,2 = 64 〈T4, T4, T5〉3,2 = 55〈T5, T5, T5〉2,3 = 6 〈T5, T5, T5〉3,3 = 12 〈T5, T5, T5〉4,3 = 6

so the quantum multiplication table is as below.

T1 ? T1 = T3 + (−18T3 + 6T4)q1(1− q1)−1 + 4q1q2 + 4q21q2T1 ? T2 = T4 + 4q1q2 + 2q21q2T1 ? T3 = (2T1 + T2)q1q2 + (−4T1 + 8T2)q21q2T1 ? T4 = T5 + (4T1 + 5T2)q1q2 + (−10T1 + 20T2)q21q2T1 ? T5 = (−8T3 + 4T4)q1q2 + (−10T3 + 4T4)q21q2 + q1q

22 + 8q21q

22 + 3q31q

22

T2 ? T2 = −6T3 + 3T4 + q2 + 4q1q2 + q21q2T2 ? T3 = T5 + (2T1 + T2)q1q2 + (−2T1 + 4T2)q21q2T2 ? T4 = 3T5 + T1q2 + (4T1 + 5T2)q1q2 + (−5T1 + 10T2)q21q2T2 ? T5 = T3q2 + (−8T3 + 4T4)q1q2 + (−5T3 + 2T4)q21q2 + 2q1q22 + 8q21q

22 + 2q31q

22

T3 ? T3 = (−2T3 + T4)q1q2 + (−20T3 + 8T4)q21q2 + 6q21q22 + 8q31q

22

T3 ? T4 = (−10T3 + 5T4)q1q2 + (−50T3 + 20T4)q21q2 + q1q22 + 20q21q

22 + 21q31q

22

T3 ? T5 = T1q1q22 + (2T1 + 6T2)q21q

22 + (−3T1 + 8T2)q31q

22

T4 ? T4 = T3q2 + (−38T3 + 19T4)q1q2 + (−50T3 + 50T4)q21q2 + 7q1q22 + 64q21q22

+55q31q22

T4 ? T5 = (4T1 + T2)q1q22 + (4T1 + 20T2)q21q22 + (−8T1 + 21T2)q31q

22

T5 ? T5 = (−T3 + T4)q1q22 + (−16T3 + 8T4)q21q22 + (−7T3 + 3T4)q31q

22

+6q21q32 + 12q31q

32 + 6q41q

32

Deforming the relations in H∗(Y,Q), we can give a presentation:

QH∗(Y ) ∼= Q[T1, T2, q2][[q1]]/〈R1, R2〉where

R1 = T 22 − 3T1T2 + 6T 2

1 − q2 − 16q1q2 − 19q21q2+18q1(1− q1)−1(T1T2 − T 2

2 + q2 − q21q2)R2 = T 3

1 + (−6T1 − T2)q1q2 − 8T2q21q2

−6q1(1− q1)−1[T 21 T2 + (−10T1 − 3T2)q1q2 + (10T1 − 24T2)q21q2].


7.11. The change of variables. Since QH∗orb([F/Z2]) has no degree 0 quantum param-

eter, we begin by setting the quantum parameter q1 to −1, in which case QH∗(Y ) isdetermined by the part of the multiplication table below

T1 ? T1 = 10T3 − 3T4 T2 ? T2 = 3T4 − 6T3 − 2q2T1 ? T2 = T4 − 2q2 T2 ? T3 = T5 + (−4T1 + 3T2)q2T1 ? T3 = (−6T1 + 7T2)q2 T2 ? T4 = 3T5 + (−8T1 + 5T2)q2T1 ? T4 = T5 + (−14T1 + 15T2)q2

and the presentation simplifies similarly:

QH∗(Y )|q1=−1∼= Q[T1, T2, q2]/〈R1, R2〉

whereR1 = 5T 2

2 − 6T1T2 + 3T 21 − 2q2

R2 = T 31 + 3T 2

1 T2 + (66T1 − 70T2)q2.

Now it is straightforward to check that QH∗orb(X )⊗QQ(i) and QH∗(Y )|q1=−1⊗QQ(i)

are isomorphic by the maps below.

S1 7→ T2

S2 7→ −i(T2 − T1)q 7→ −q2

T1 7→ S1 − iS2

T2 7→ S1

q2 7→ −qWe will further discuss this change of variables in the next section (7.12). Clearly theseare inverse maps, so all we need to do is show that they are well defined (i.e. that theykill the relations). The most difficult such computation is checking that S3

2 − 6qS2 7→ 0.To do this, use the relations T1R1, T2R1, and R2 (or the multiplication table) to expressthe degree three monomials in terms of T 2

1 T2:

T 31 = −3T 2

1 T2 + (−66T1 + 70T2)q2T1T

22 = 3T 2

1 T2 + (40T1 − 42T2)q2T 3

2 = 3T 21 T2 + (48T1 − 50T2)q2

so thati(S3

2 − 6qS2) 7→ T 31 − 3T 2

1 T2 + 3T1T22 − T 3

2 + 6q2(T2 − T1)= −3T 2

1 T2 + (−66T1 + 70T2)q2−3T 2

1 T2

+9T 21 T2 + (120T1 − 126T2)q2

−3T 21 T2 + (−48T1 + 50T2)q2 + (−6T1 + 6T2)q2

= 0.

The other checks are similar, though less difficult.

7.12. Higher point invariants of X . In this section we show that the genus zeroGromov-Witten invariants of X are determined by the 3-point invariants that we alreadycomputed (in fact we will show that only the 2-point invariants are needed) together withthe degree zero invariants

〈S2, . . . , S2〉0,which can be computed as in Section 5. In what follows, by a divisor, we will mean a“non-twisted” divisor: a cohomology class δ ∈ H2(X ) ⊆ H2

orb(X ). Recall that the usualWDVV equation 36 holds for orbifolds without modification [5], but the Point Mappingand Divisor Axioms do not (hence Equations 37 and 38 are not quite right for orbifolds,though the extra terms that show up in Equations 37 and 38 occur when 0 < |V | < n


and hence only involve invariants of lower point number). This is because both of theseaxioms are proved (for smooth varieties) using the forgetful stabilization map

M0;n+1(X,β) →M0;n(X,β)

(which exists when n ≥ 3 or β 6= 0). However, when X is an orbifold (say X = [M/Z2])then, although there is a forgetful map

M0;r,u+1(X , β) →M0;r,u(X , β),

(at least when r + u ≥ 3 or β 6= 0) there is no forgetful map

M0;r+1,u(X , β) →M0;r,u(X , β).

Now we recall a simple reconstruction theorem for genus zero Gromov-Witten invari-ants, which is particularly relevant for orbifolds. This is essentially the natural statementof Kontsevich’s reconstruction theorem when the cohomology ring is not generated bydivisors; it appears in [55] in essentially the same form as below (we include the proof hereas well since it is so simple). Let H∗

orb(X )div denote the subring of H∗orb(X ) generated

by divisors (in the sense above).

Lemma 7.12.1. Suppose that θ1, . . . , θN ∈ H∗orb(X ) generate H∗

orb(X ) as a module overH∗

orb(X )div. Then the genus zero Gromov-Witten invariants of X are determined bylinearity, the WDVV equations, the Divisor Axiom, the 2-point invariants, the 3-pointdegree 0 invariants, and the invariants of the form

〈ηθi1 , θi2 , . . . , θik〉β,where η ∈ H∗

orb(X )div.

Proof. We say 〈α1, . . . , αn〉β1 is lower than 〈α′1, . . . , α′m〉β2 if n ≤ m, β1 ≤ β2, and one of

these inequalities is strict. By the hypothesis on the θi and linearity it suffices to showthat an invariant of the form

〈δ1,i1δ1,i2 · · · δ1,im1θj1 , δ2,i1δ2,i2 · · · δ2,im2

θj2 , . . . , δn,i1δn,i2 · · · δn,imnθjn〉β,

where the δi,j are various divisors depending on the pair (i, j) (some mj may be zero),is determined by the data mentioned above. We may assume by induction that all lowerinvariants are determined by such data and that n ≥ 3. Notice that if δ is a divisor and

〈α1, . . . , αi−1, δαi, αi+1, . . . , αn〉β

is an invariant with n ≥ 3, then applying WDVV (36) with φ1 = αi, φ2 = δ, φ3 = α1, andφ4, γ1, . . . , γn−3 equal to α2, . . . , αi−1, αi+1, . . . , αn expresses the above invariant in termsof lower invariants, the invariant

〈δα1, α2, . . . , αn〉β,and two other invariants which are immediately reduced to lower invariants by the DivisorAxiom. Applying this repeatedly we can move all the δi,j to the first insertion to establishthe lemma. ¤

When X = [F/Z2], θ1 = S2 satisfies the hypotheses on the θi above. In H∗orb(X ),

S1S2 = 4S4, Sn1 S2 = 0 for n > 1, and the invariants

〈S4, S2, S2, . . . , S2〉d


all vanish for dimension reasons, so the genus 0 Gromov-Witten theory is determined bypreviously computed invariants and the invariants 〈S⊗2g+2

2 〉0. As discussed in Section 5, thecorresponding moduli space M0;2g+2,0(X , 0) is just C ×M0;2g+2,0(BZ2) and the virtualfundamental class is dual to the Euler class of

π∗1NC/F ⊗ π∗2E∨,

where E is the Hodge bundle on M0;2g+2,0(BZ2). Let λi := ci(E). Since p1|C = p2|C = 2,and c1(TF ) = 2p1 + 2p2, the SES

0 → TC ∼= TP1 → TF |C → NC/F → 0

shows that c1(NC/F ) = 6. Thus we compute

〈S⊗2g+22 〉0 =

∫

M0;2g+2,0(X ,0)e(Obstruction Bundle)

=∫

C×M0;2g+2,0(BZ2)c2g(π∗1NC/F ⊗ π∗2E∨)

= −6∫

M0;2g+2,0(BZ2)λgλg−1

These integrals were computed in [29] and in Section 4.

To complete the proof of the Crepant Resolution Conjecture, we determine the changeof variables to use by setting q = −q2 = 0 in Section 7.11 and determine the matrix givingthe change of variables Si → Ti. For example, S4 = 1

4S1 ? S2 and since we haveS1 7→ T2, S2 7→ −i(T2 − T1) we want

S4 7→ 14T2(iT1 − iT2)

= i4T1T2 − i

4T22

= i4T4 − i

4(3T1T2 − 6T 21 )

= 3i2 T3 − i

2T4.

Carrying out similar computations we get the change of variables matrix:

1i

1 −i−6 3i/23 −i/2

3


We have shown that the potential functions of Y and X are given as below (neglectingterms of degree ≥ 4 in the si and ti, as well as “unstable terms” of degree < 3).

ΦY (t0, . . . , t5, q1, q2) =t20t52

+ t0t1t4 + t0t2t3 + t0t2t4 +t21t22

+3t1t22

2+t322

+t31q1

1− q1

+t22t5q2

2+t2t

24q22

+ 2t21t5q1q2 + 4t1t2t5q1q2 +t1t

23q1q22

+5t1t3t4q1q2 +19t1t24q1q2

2+ 2t22t5q1q2 +

t2t23q1q22

+ 5t2t3t4q1q2

+19t2t24q1q2

2+ 2t21t5q

21q2 + 2t1t2t5q21q2 + 4t1t23q

21q2

+20t1t3t4q21q2 + 25t1t24q21q2 +

t22t5q21q2

2+ 2t2t23q

21q2 + 10t2t3t4q21q2

+25t2t24q

21q2

2+t1t

25q1q

22

2+ 4t1t25q

21q

22 +

3t1t25q31q

22

2+t2t25q1q

22 + 4t2t25q

21q

22 + t2t

25q

31q

22 + 3t23t5q

21q

22 + 4t23t5q

31q

22

t3t4t5q1q22 + 20t3t4t5q21q

22 + 21t3t4t5q31q

22 +

7t24t5q1q22

2

+32t24t5q21q

22 +

55t24t5q31q

22

2+ t35q

21q

32 + 2t35q

31q

32 + t35q

41q

32 +O(4)

ΦX (s0, . . . , s5, q) =3s20s5

2+ 3s0s1s3 +

s0s2s42

+s312

+ s1s22 +

9s1s23q2

+ 3s21s5q

+27s23s5q2 + 18s1s25q

2 +s1s

24q

2+ 3s2s3s4q + 3s22s5q +

3s24s5q2

2+O(4)

It is easy (with a computer, say) to check that

ΦY (s0, is2, s1 − is2,−6s3 +3i2s4, 3s3 − i

2s4, 3s5,−1,−q) = ΦX (s0, . . . , s5, q).

Since the change of variables is linear and respects the metrics, it preserves WDVV equa-tions, so the entire potentials will agree (because of Lemma 7.12.1) if the coefficients ofsn2 (with no powers of q) agree. In particular, it will be sufficient to show that the full

potential functions agree (up to unstable terms) under the above change of variables (an-alytically continuing to q1 = −1) when we set q = s0 = s3 = s4 = s5 = 0. Notice that,because of the Divisor Axiom and the fact that T2 evaluates 0 on a homology class of theform (l, 0), the dependence on t2 in the potential function of Y is purely classical whenwe set q2 = 0. Also notice that the coefficient of tn1 (when q2 = 0) is determined by theinvariant 〈T1, . . . , T1〉d,0, which reduces by the Divisor Axiom to our previous computationof 〈〉d,0. Putting these observations together, we have

ΦY (0, t1, t2, 0, 0, 0, q1, 0) =t21t22

+3t1t22

2+t322

+ 6∑

d≥1

1d3edt1qd

1

and, using the computation above,

FX (0, s1, s2, 0, 0, 0, 0) =s312

+ s1s22 − 6

∑

g≥1

1(2g + 2)!

(∫

M0;2g+2,0(BZ2)λgλg−1

)s2g+22 ,


so

ΦY (0, is2, s1 − is2, 0, 0, 0, 0) =s312

+ s1s22 + 6

− is

32

12+

∑

d≥1

1d3edis2qd

1

.

Clearly all the terms involving s1 match up, and the fact that the third derivatives withrespect to s2 agree after analytically continuing to q1 = −1 is the same computation wedid in Section 5 (both of these third partials are equal to −3 tan(s2/2)).

Using the cohomology calculations from Section 7.2 we see that our change of variablesis the same as the one in Theorem 5.0.4. For example, we have

H∗(F/Z2,Q) ∼= Q[(p1 + p2)]/〈(p1 + p2)4〉with the pullback by the resolution map r∗ : H∗(F/Z2) → H∗(Y ) taking (p1 + p2) toT2 ∈ H2(Y ). In H∗(X ) ⊂ H∗

orb(X ) we have S1 = (p1 + p2) and indeed, our change ofvariables is given by

S1 = (p1 + p2) 7→ T2 = r∗(p1 + p2)S3 = (p1 + p2)2 7→ −6T3 + 3T4 = T 2

2 = r∗(p1 + p2)2

S5 = (p1 + p2)2 7→ 3T5 = T 32 = r∗(p1 + p2)3.

In Section 7.2 we showed that the Poincare dual of the exceptional divisor of r is

2H = 2r∗(p1 + p2)− 2T1 = 2T2 − 2T1

and our change of variables maps S2 = 1 ∈ H0(C) ⊂ H2orb(X ) to

−i(T2 − T1) = − i2(P.D. [E]) = − i

2j∗1,

where j : E → Y is the inclusion of the exceptional divisor and 1 ∈ H∗(E).


Index of Notation

Notation Meaning Section

k algebraically closed field of characteristic zero 2S category of schemes over Spec k 2X a Deligne-Mumford stack, usually smooth 2X the coarse moduli space of X 2µr group scheme of rth roots of unity in k 2.1Mg;n,r(X , d) stack of degree d twisted stable maps to X 2.1Mg,n,r Artin stack of twisted curves 2.3IX rigidified cyclotomic inertia stack of X 2.2.1〈α〉dX Gromov-Witten invariant 2.4H∗

orb(X ) orbifold cohomology ring of X 2.4r : Y → X a crepant resolution of X 3M smooth projective variety, typically with Z2 action 3C fixed locus of the Z2 action on M 3.0.2E exceptional divisor of the resolution r 3ψi cotangent line class 4λi ith Chern class of the Hodge bundle 4E Hodge bundle 4Li cotangent line bundle 4Ψ Perroni’s cohomology isomorphism 5.0.3Li line bundle on a curve 6li degree of Li 6X0, . . . , X3 basis for H∗

orb(X ) 6.1Md,g,n torus fixed stable maps to [L1 ⊕ L2/Z2] 6.1C, C, C various universal curves over Md,g,n 6.1f, f , f various universal maps over Md,g,n 6.1Obi

d,g obstruction class from Li 6.1gX orbifold Poincare pairing matrix for X 6.1gabX (a, b) entry of the modified inverse Poincare pairing for X 6.2Z generating series for non-equivariant invariants of X 6.3Y0, . . . , Y3 additive basis for the cohomology of P(L1 ⊕ L2) 6.4gY Poincare pairing matrix for Y 6.4gabY (a, b) entry of the modified inverse Poincare pairing for Y 6.4.3W generating series for non-equivariant invariants of Y 6.5S0, . . . , S5 additive basis for H∗

orb([F/Z2]) 7.3T0, . . . , T5 additive basis for H∗(Y ) 7.9H∗

orb(X )div subring of H∗orb(X ) generated by H2(X ) ⊆ H∗

orb(X ) 7.12


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hyperelliptic gromov-witten theory · we prove a “degree zero” version of this conjecture,...

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