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ContentsMacPherson’s natural transformation

‘Unexpected’ applicationsReferences

(Unexpected) applications of characteristic classesfor singular varieties

Paolo Aluffi

April 3, 2009

Paolo Aluffi CSM, HEP, etc.



1 MacPherson’s natural transformationGood invariants of algebraic varietiesExample: constructible functionsCharacteristic classes for (possibly) singular varietiesExplicit constructionChern-Fulton class and CSM class

2 ‘Unexpected’ applicationsEffective computationsBirational invariance of Chern classes#vir (Donaldson-Thomas-Behrend)Chern class identities from tadpole matchingGraph hypersurfaces & Feynman integrals

3 References




Good invariants of algebraic varietiesExample: constructible functionsCharacteristic classes for (possibly) singular varietiesExplicit constructionChern-Fulton class and CSM class

Working over C, or any algebraically closed field of char. 0.

X variety good invariants associated with X .

number?

function?

Chow class?

. . .

What do I mean by good?

additive w.r.t. disjoint union

(multiplicative w.r.t. products)

(more? controlled behavior under morphisms)





Prototype invariant: Euler characteristicUniversal Euler characteristic: class in the Grothendieck ring ofvarieties.

Generators (as abelian group): isomorphism classes of varieties

Relations: [X ] = [Z ] + [X r Z ] for every X , Z → X closed;

multiplication: [X ] · [Y ] = [X × Y ].

Additive invariants with fixed target factor through this ring.Examples: topological Euler characteristic, Hodgepolynomial. . . number of points over finite fields. . . class in theGrothendieck group of motives. . .But there is more.





X F (X )= ∑

V⊆X mV 11V , constructible functions

This is a covariant functor:

f : X → Y , V ⊆ X , p ∈ Y ;f∗(11V ) =χtop(f

−1(p) ∩ V ).

Q. Is there a ‘good’ invariant in F (X )?A. Yes. Choose 11X ∈ F (X ), for any X .

Not impressed? Note that the class [X ] in the Grothendieck ring ofvarieties may be read off 11X ∈ F (X ), but not conversely. So this isa finer invariant than the class in the Grothendieck ring.For example, χ(X ) may be compatibly associated with 11X .





Important remark:

If X is nonsingular, and nothing special is known about X ,then 11X is the ‘only’ available constructible function on X .

If X is singular, many other natural choices:

11X ;multX (multiplicity);EuX (‘local Euler obstruction’);νX (Behrend);. . .σX , if X is a hypersurface in a nonsingular variety (‘Verdierspecialization’);more? (stringy. . . )

All agree with 11X when X is nonsingular.





Summary: The constructible function functor carries stronger‘good’ invariants than the Grothendieck ring of varieties.

Q.: Does Chow carry an interesting ‘good’ invariant?A!: It inherits one from constructible functions.

Hint: The degree of such an invariant ‘must be’ the Eulercharacteristic. Thus: if X is compact, nonsingular, then the ‘only’reasonable candidate is c(TX ) ∩ [X ].

Grothendieck-Deligne (ca. 1968): Does there exist a naturaltransformation c∗ : F A∗, such that

c∗(11X ) = c(TX ) ∩ [X ]

if X is nonsingular?





Grothendieck-Deligne (ca. 1968): Does there exist a naturaltransformation c∗ : F A∗, such that c∗(11X ) = c(TX ) ∩ [X ] if Xis nonsingular?

MacPherson: Yes! Explicit construction.(Chern classes for singular algebraic varieties. Ann. of Math. 100,423–432 (1974).)

c∗(11X ): Exercise, degree = χ(X ).This class agrees with one previously introduced by Marie-HeleneSchwartz. (Brasselet-Schwartz, ca. 1980.)

Definition

cSM(X ) := c∗(11X ) is the Chern-Schwartz-MacPherson (CSM)class of X .





Definition

cSM(X ) := c∗(11X )

“If X is smooth, and nothing special is known about X , then11X is the only available constructible function on X .” if X is smooth, cSM(X ) = c(TX ) ∩ [X ] is the onlyinteresting class one may construct this way.

“If X is singular, many possibly choices.”

c∗(EuX ) =: cMa(X ) Chern-Mather class. (MacPherson)c∗(σX ) =: cF(X ) Chern-Fulton class, if X is a hypersurface.(Fulton, IT, Example 4.2.6.)c∗(νX ) cwMa(X ) weighted Chern-Mather class (up tosigns.)(—, 2000.)More? (Stringy Chern class. . . ) E.g., c∗(multX ) ??





ϕ ∈ F (X ) c∗(ϕ) ∈ A∗XWrite ϕ =

∑V mV 11V , for V nonsingular (of course, possibly

noncompact). By linearity, enough to define c∗(11V ) ∈ A∗X .

W

!!BBB

BBBB

Bw

((PPPPPPPPPPPPPPP

V?

OO

// V // X

W := resolution of singularities of V .

D := W r V , assume divisor with SNC.

Definition

c∗(11V ) = w∗(c(ΩW (log D)∨) ∩ [W ])





Definition

c∗(ϕ) :=∑

V mV c∗(11V )

Theorem (—, 2006)

This is independent of all choices, and agrees with MacPherson’sdefinition in terms of Chern-Mather classes and local Eulerobstructions.

Two proofs:

Using MacPherson’s natural transformation, easy exercise.

Not using MacPherson’s natural transformation, prove directlythat c∗ satisfies the Grothendieck-Deligne requirements.

Useful side-product: covariance with respect to not necessarilyproper morphisms, for an ‘enlarged’ Chow functor.





X hypersurface in a nonsingular variety M

cF(X ) := c∗(σX ), where σX = Verdier specialization function.

More concretely: X = any complete intersection in a nonsingularvariety M. Then cF(X ) := c(TM) ∩ (c(NXM)−1 ∩ [X ]).In general:

Definition

cF(X ) := c(TM) ∩ s(X ,M)

where s(X ,M) = Segre class of X in M. The definition isindependent of M (Fulton, Example 4.2.6).

Morally, cF(X ) is ‘the Chern class of a smoothening of X ’; in somecases this statement can be made precise (Fantechi-Gottsche 2007,Theorem 4.15).





‘Comparison theorem’

If X is a hypersurface in a nonsingular variety M, there is anexplicit relationship between cSM(X ) and cF(X ).

Formula for cF(X (m)), where X (m) is obtained by ‘thickening’X along its singularity subscheme Y . (Y defined by partials oflocal equation for X )

cF(X (m)) is a polynomial in m, so it makes sense fornegative m.

Theorem (—, 1994, 1999)

cSM(X ) = cF(X (−1))

Equivalently: explicit formula for cSM(X ) in terms of cF(X ),c(O(X )), and s(Y ,M).





In the stacky world?

X : with Gorenstein quotient singularitiesX : smooth DM stack with coarse moduli space XIX : inertia stack

Theorem (Tseng, 2007)

cSM(X ) = push-forward to X of c(TIX )

(What about cF(X )?)




Effective computationsBirational invariance of Chern classes#vir (Donaldson-Thomas-Behrend)Chern class identities from tadpole matchingGraph hypersurfaces & Feynman integrals

http://www.math.fsu.edu/ aluffi/CSM/CSMexamples.html

i19 : CSM threecoplanarlines

3 2

Chern-Schwartz-MacPherson class : 4H + 3H

The Euler characteristic is the coefficient of H3, that is (of course) 4.

For an example of two nonisomorphic schemes with the same Fulton class, but different Chern-Schwartz-MacPherson classes, consider

the union of three planes in P3, vs. a nonsingular cubic surface:

i20 : use ringP3;threeplanes=ideal(x*y*z);cubic=ideal(x^3+y^3+z^3+w^3);

o21 : Ideal of ringP3

o22 : Ideal of ringP3

i23 : CF threeplanes; CF cubic;

3 2

Fulton class : 9H + 3H + 3H

3 2

Fulton class : 9H + 3H + 3H

i25 : CSM threeplanes; CSM cubic;

3 2

Chern-Schwartz-MacPherson class : 4H + 6H + 3H

3 2

Chern-Schwartz-MacPherson class : 9H + 3H + 3H

The Fulton and Chern-Schwartz-MacPherson classes agree for the second scheme since it is nonsingular. The Euler characteristic is

computed to 9, as it should. To compute the Euler characteristic of the (complex) affine nonsingular cubic surface x3+y3+z3=1:

i27 : use QQ[x,y,z]; euleraffine ideal(x^3+y^3+z^3-1)

o28 = 9

This is also 9, as the cubic curve `at infinity' is nonsingular. Not so for the affine nonsingular cubic surface x3+y3+z2=1, and the Eulercharacteristic drops accordingly:

i29 : euleraffine ideal(x^3+y^3+z^2-1)

(—, 2003): Macaulay2 code to compute the (push-forward toambient projective space of the) CSM class and Fulton-Chern classof a subscheme of Pn, given the ideal.

Term of dim 0 in CSM computes the Euler characteristic.





How does it work?

Functoriality of CSM classes ‘inclusion-exclusion’

reduce to the case of a hypersurface

‘Comparison theorem’ reduce to computation of Fultonclass of hypersurfaces (easy) and of Segre classes (not easy!)

Compute Segre classes: Rees algebra, intersections with‘general’ hyperplanes, degree computations.

If X is nonsingular, recent work of di Rocco et al.

CSM routine works for arbitrary X , and uses classes for singularvarieties even to compute the class if X is nonsingular.

(This is what is ‘unexpected’.)





X , Y : K -equivalent nonsingular complex varieties

VπX

πY

999

9999

X Y

πX , πY proper birational, KπX∼ KπY

.Example: X , Y birational Calabi-Yau.

Theorem (—, 2004)

∃C ∈ A∗V such that

πX∗(C ) = c(TX ) ∩ [X ] , πY ∗(C ) = c(TY ) ∩ [Y ]

Corollary: Batyrev’s observation that χ(X ) = χ(Y ).





Two proofs:

Direct, using factorization theorem of ℵKMW.

Not-so-direct: mimic motivic integration, prove for CSMclasses.

Analog of motivic integration for Chow:X : variety; D: divisor; S : subvariety

∫

S11(D) dcX

This lives in a limit of Chow groups; it has a value for each Vmapping properly and birationally to X .(For the purpose of this talk, put it in A∗X .)

Main property: change-of-variables





Change-of-variables: ρ : V → X proper birational =⇒∫S

11(D) dcX = ρ∗

∫ρ−1(S)

11(ρ∗D + Kρ) dcV

Theorem (—, 2005)

If X is smooth, S ⊆ X, then cSM(S) =∫S 11(0) dcX

X , Y K -equivalent:

Change of variables in motivic integration =⇒ [X ] = [Y ] ina localization of the Grothendieck ring of varieties

Change of variables in Chow integ. =⇒ ‘cSM(X ) = cSM(Y )’in limit of Chow, i.e., the statement given earlier.





Example: c i1 · cn−i is an invariant of K -equivalent normal varieties

of dimension n.

More generally: explicit formulas relating Chern classes ofbirational varieties, (messy) identities for Chern numbers.





Formula as above:

S ⊆ X , X nonsingular =⇒∫S 11(0) dcX = cSM(S)

Alternative: X possibly singular, S = X =⇒∫X 11(0) dcX =?

Theorem (—, 2006)

Degree of this class = Batyrev’s stringy Euler number

Proof: Relate directly to motivic integration. (‘User-friendlyformula’ in Craw.)

Definition∫X 11(0) dcX =: ‘stringy Chern class’ of X

May be written as c∗ of a ‘stringy constructible function’: seedeFernex-Lupercio-Nevins-Uribe, article on Advances. (IncludesMcKay correspondence.)





Remarks:∫X 11(0) dcX a well-defined class for every variety mapping

properly birationally to X . (Same with∫S 11(0) dcX .)

In some interesting situations, stringy = Chern-Mather.E.g.: for Schubert varieties (B. Jones, 2008).

X with Gorenstein quotient singularities; X smooth DM stackwith coarse moduli space X :

Theorem (Tseng, 2007)

Stringy Chern class of X = push-forward to X of c(TIIX )





Z : Calabi-Yau 3-foldY = Z [n] = HilbnZ= Crit F , F suitable Chern-Simons functional.

Thomas: #virY .

Theorem (Behrend, 2005)

#virY = χ(Y , νY )

νY = Behrend’s constructible function.

Corollary (Proposition 1.12 in Behrend): #virY = ±∫

cwMa(Y )

cwMa(Y ) = ‘weighted Chern-Mather class’ (—, 2000), a certainlinear combination of Chern-Mather classes.

(Note: sensitive to scheme structure, like cF and unlike cSM. . . )





Example: Y smooth #virY = ±∫

c(TY ) ∩ [Y ] = ±χ(Y ).

In general? Behrend et al.: in DT case, Y = locally Crit f ,f holomorphic on finitely dimensional M.

Prototype (global) situation:

X = hypersurface in nonsingular variety M

X = f = 0, f section of line bundle LY = Crit f = ‘singularity subscheme’ of X .

Then νY (p) = ±(1− χ(Xp)), Xp = Milnor fiber at p.(Ref.: Parusinski-Pragacz, JAG 2001).

(There are problems localizing this formula?)





In the same situation, cwMa(Y ) =?

Two formulas from 2000.

(Class version of Parusinski-Pragacz.) ι : Y → X

Context: ±(cF(X )− cSM(X )) =: Milnor class of X .

X compact:∫

(cF(X )− cSM(X )) = Parusinski-Milnor # of X .(dim Y = 0 sum of ordinary Milnor numbers)

Theorem

ι∗cwMa(Y ) = ±c(L) ∩ (cF(X )− cSM(X ))

Hence: Behrend’s #virY = ±∫

c(L) ∩ (cF(X )− cSM(X )).

Can one make sense of this when dim X = ∞??





Recall cF(Y ) = c(TM) ∩ s(Y ,M).

Theorem

cwMa(Y ) = (−1)dimY (c(T ∗M ⊗ L) ∩ s(Y ,M))∨L

Funny notation ‘∨L’: some other time.

Hence: #virY = ±∫ (

c(T∗M⊗L)c(TM) ∩ cF (Y )

)∨L

.

Is this intrinsic enough to localize?

(L|Y = Ext1(ΩX ,OX )|Y )





(joint work with Mboyo Esole)

D-branes

‘Most important development in string theory’ (Aspinwall)

‘Dirichlet boundary conditions’ for ends of open strings

First approximation: subvarieties endowed with vector bundles

They carry charge, which may be computed by RR formulas

Dualities may be used to relate number and charges ofD-branes in apparently different situations

identities for Euler characteristics of relevant loci





Sen’s weak coupling limit of F-theory as a type IIB orientifold:

B: nonsingular compact algebraic 3-fold

Y → B: E8-elliptic fibration; Y a CY

Degenerate Y (after Sen); find surfaces O, D supportinglimits of discriminants

X := double cover of B, ramified along O; also CY

D := inverse image of D in X

‘Compatibility of tadpole conditions’:

2χ(Y )?= χ(D) + 4χ(O)

Esole: this is a non-negotiable identity.





Example:

B = P3

Y : y2 = z3 + fz + g in P1,1,1,1,8,12, deg y = 12, deg z = 8,deg f = 16, deg g = 24

. . .

Non-negotiable identity: 46,656!= 55,360

???

Key:

Charges were computed as if the support D of the relevantD-brane were nonsingular (HRR).Discrepancy is due to singularities of D.





Modulo white lies:

Theorem (—, Esole, 2009)

Identity holds for analogue of stringy χ;

It generalizes to stringy Chern classes;

Chern class identity holds regardless of CY hypotheses, and inall dimensions.

(Here, ‘stringy’ is not quite the stringy notion reviewed earlier)





One ingredient:

Y , B smooth; ϕ : Y → B elliptic fibration, Weierstrass equationy2 = z3 + fzx2 + gx3, with f , g sufficiently general.G : g = 0; ι : G → B

Theorem (2009)

ϕ∗c(TY ) ∩ [Y ] = 2ι∗c(TG ) ∩ [G ]

Proof: Calculus of constructible functions reduce tocomputation of cSM for the discriminant; use ‘comparisontheorem’.(Again, singular tools are used to answer nonsingular question)





If

dim Y = 4;

Y is a CY;

only look at term of dim 0;

then get

χ(Y ) = 360c31 + 12c1c2

where ci = ci (TB).This is sometimes referred to as the Sethi-Vafa-Witten formula,from Nucl. Phys. B 480 (1996) p. 213.





Perturbative QFT Feynman integral computationsExtensive numerical evidence: multiple zeta values(Broadhurst-Kreimer)Hard to give a precise statement, as integrals typically diverge.Γ: graph; p: ‘momenta’ attached to external edges

U(Γ, p) =Γ(n − D`/2)

(4π)`D/2

∫[0,1]n

δ(1−∑

i ti )VΓ(t, p)D`/2−n

ΨΓ(t)D/2dt1 · · · dtn.

n = # internal edges

D = spacetime dimension

` = b1(Γ) = # loops

VΓ(t, p) = a rational function

ψΓ(t)= a polynomial of degree ` determined by the graph.





Ignore most of this!

U(Γ, p) = an integral of a form defined over the complement of ahypersurface XΓ: ψΓ = 0.

Kontsevich: BK evidence may be explained if graph hypersurfacesXΓ determine mixed-Tate motives.

Belkale-Brosnan: not true. Graph hypersurfaces generate theGrothendieck ring of varieties! (But the proof is non-constructive.)

Program:

Analyze classes of graphs, attempt to estimate ‘complexity’ ofXΓ in the Grothendieck ring.Note: ‘Simple’ in Grothendieck ring only if ‘very’ singular.‘Quantify’ singularity: cF(XΓ)− cSM(XΓ) =?.Tools to compute cSM(XΓ) usually suffice in order to computeclass in Grothendieck ring.





Γ: graph; one variable te for each edge e(Usually assume Γ is connected and 1–PI: it cannot bedisconnected by removing a single edge.)

Definition

ΨΓ(t) =∑T⊆Γ

∏e /∈E(T )

te

where the sum is over all the spanning trees T of Γ.

# of variables = # (internal) edges; degree = # loopsExample: Γ=n-sided polygon ψΓ = t1 + t2 + · · ·+ tnExample: ‘banana graphs’: two vertices, n parallel edges

ψΓ = t1 · · · tn(

1t1

+ · · ·+ 1tn

).





ψΓ = 0: hypersurface XΓ ⊆ Pn−1 (or XΓ ⊆ An); deg XΓ = `.

Theorem (—, Marcolli, 2008)

Explicit computation of [XΓ] ∈ Grothendieck ring, and cSM(XΓ),for Γ = all banana graphs

Corollary: χ(XΓ) = n + (−1)n for n ≥ 3.

Remark: In particular, χ(XΓ) > 0 for all banana graphs.In fact, cSM(XΓ) is effective for banana graphs.Computer calculations (J. Stryker): we do not know of a singlegraph Γ for which cSM(XΓ) is not effective.





Proof of the theorem:

If Γ is any planar graph, can relate XΓ to XΓ∨ , where Γ∨ is the dualgraph: they correspond to each other via a Cremonatransformation of Pn−1.For Γ = banana graphs, Γ∨ are polygons, computation can becarried out explicitly. (Calculus of constructible functions, andlemma on CSM classes via ‘adapted blow-ups’.)

Remark: More generally, one expect [XΓ]± [XΓ∨ ] to be ‘easier’ thaneither class. Bloch, 2008: computation of

∑[XΓ], Γ connected

graph with N vertices (with automorphism factor); it is MT.





Reason why Γ assumed to be connected, 1–PI:Integrals U(Γ, p) are multiplicative on disjoint unions of graphs. IfΓ = Γ1 q Γ2, then

U(Γ, p) = U(Γ1, p1)U(Γ2, p2)

If Γ is obtained by joining Γ1, Γ2 by an edge, multiply product by a‘propagator’ term.

FEYNMAN RULES!

With Marcolli: ‘Algebro-geometric Feynman rules’(I vetoed ‘Feynman rules in algebraic geometry’)





The following amazing recipe is part of a larger picture:

Γ: finite graph (may be non-connected, non-1-PI. . . ), n edges

XΓ: corr. hypersurface in An; view as locally closed in Pn

c∗(11XΓ) = a0[P0] + · · ·+ an[Pn]

Define GΓ(T ) = a0 + a1T + · · ·+ anTn

Define CΓ(T ) = (T + 1)n − GΓ(T )

Example: Γ = banana graph CΓ(T ) = T (T − 1)n−1 + nT n−1

Remarks:

Coefficient of T n−1 in CΓ(T ) equals n − `.

C ′Γ(0) = χ(Pn−1 r XΓ).





Theorem (—, Marcolli, 2008)

The invariant CΓ(T ) obeys the Feynman rules, with inversepropagator (T + 1).

Proof:Show that Feynman rules correspond to homomorphisms from a‘Grothendieck ring’ of conical immersed subvarieties of An.The function GΓ(T ) is such a homomorphism.(This does not really have directly to do with graphs.)Proof of this fact: study cSM classes of joins in projectivespace.





More recent work with Marcolli: a possible approach to explainingthe BK evidence.

Idea: Transfer the integral computation to a fixed variety D`

(for given number ` of loops) for all graphs with ` loops, theFeynman integral is a period of a fixed D` relative to a locus S`

supported on strata of a fixed normal crossing divisor.

This reduces the question to ‘linear algebra’: describe a variety offrames (v1, . . . , v`) with v1 ∈ V1, . . . , v` ∈ V`, where V1, . . .V` are(arbitrary) subspaces of a fixed vector space.

But this is material for another talk.





SUMMARY:

Chern classes of singular varieties show up in unexpectedplaces.

Often, results about Chern classes for nonsingular varieties arestreamlined and clarified by the use of tools from the theory ofChern classes for singular varieties.




REFERENCES:

MacPherson’s and Fulton’s Chern classes of hypersurfaces. Internat.Math. Res. Notices (1994), 455–465.

Chern classes for singular hypersurfaces. Trans. Amer. Math. Soc.351 (1999), no. 10, 3989–4026.

Weighted Chern-Mather classes and Milnor classes of hypersurfaces.Singularities—Sapporo 1998, 1–20, Adv. Stud. Pure Math., 29(2000).

Computing characteristic classes of projective schemes. J. SymbolicComput. 35 (2003), no. 1, 3–19.

Chern classes of birational varieties. IMRN 63 (2004) 3367-3377.




Classes de Chern des varietes singulieres, revisitees. C. R. Math.Acad. Sci. Paris 342 (2006), no. 6, 405–410.

Limits of Chow groups, and a new construction ofChern-Schwartz-MacPherson classes. Pure Appl. Math. Q.,(MacPherson Volume 2) (2006) 915–941.

Chern class identities from tadpole matching in type IIB andF-theory. (with Mboyo Esole) J. High Energy Phys. 2009, no. 3,032, 29 pp.

Feynman motives of banana graphs. (with Matilde Marcolli)arXiv:0807.1690 (to appear in CNTP).

Algebro-Geometric Feynman rules. (with Matilde Marcolli)arXiv:0811.2514.


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