thed-criticalstructure onthequotschemeofpointsofacalabi

arX

iv:2

106.

1613

3v1

[m

ath.

AG

] 3

0 Ju

n 20

21

The d-critical structure

on the Quot scheme of points of a Calabi–Yau 3-fold

ANDREA T. RICOLFI AND MICHAIL SAVVAS

ABSTRACT. The Artin stackMn of 0-dimensional sheaves of length n onA3 carries two natural

d-critical structures in the sense of Joyce. One comes from its description as a quotient stack

[crit( fn )/GLn ], another comes from derived deformation theory of sheaves. We show that these

d-critical structures agree. We use this result to prove the analogous statement for the Quot

scheme of points QuotA3 (O ⊕r , n ) = crit( fr,n ), which is a global critical locus for every r > 0, and

also carries a derived-in-flavour d-critical structure besides the one induced by the potential

fr,n . Again, we show these two d-critical structures agree. Moreover, we prove that they locally

model the d-critical structure on QuotX (F, n ), where F is a locally free sheaf of rank r on a

projective Calabi–Yau 3-fold X .

Finally, we prove that the perfect obstruction theory on Hilbn A3 = crit( f1,n ) induced by the

Atiyah class of the universal ideal agrees with the critical obstruction theory induced by the

Hessian of the potential f1,n .

CONTENTS

0. Introduction 1

1. Background material 5

2. Proof of Theorem A 9

3. The d-critical structure(s) on Mn 11

4. The d-critical structure(s) on QuotA3 (O ⊕r , n ) 23

5. The case of a compact Calabi–Yau 3-fold 27

6. The special case of the Hilbert scheme 32

References 36

0. INTRODUCTION

0.1. Overview. In [20], Joyce introduced d-critical structures on schemes and Artin stacks,

both in algebraic and analytic language. Roughly speaking, a d-critical scheme is a scheme X

suitably covered by ‘d-critical charts’, i.e. schemes of the form crit( f ) ,→ V , where f ∈ O (V ) is

a regular function on a smooth scheme V and crit( f ) denotes the scheme-theoretic vanish-

ing locus of d f ∈ H0(V ,ΩV ). The compatibility between d-critical charts is governed by the

behaviour of a section s of a canonically defined sheaf of C-vector spaces S0X living over X .

Such a well-behaving section is called a ‘d-critical structure’ on X (see Section 1.2 for more

details). We use the notation (X , s ) to represent a d-critical scheme.

Joyce’s d-critical loci play a central role in categorified Donaldson–Thomas theory: the

compatibility between d-critical charts encoded in their structure is crucial to associate to1

http://arxiv.org/abs/2106.16133v1

2 ANDREA T. RICOLFI AND MICHAIL SAVVAS

a moduli space M of sheaves on a Calabi–Yau 3-fold a canonical1 perverse sheaf ΦM (the so-

called DT sheaf ), as well as a canonical virtual motive ΦmotM (the so-called motivic DT invari-

ant).

Any d-critical chart crit( f ) ,→ V has a canonical symmetric perfect obstruction theory in

the sense of Behrend–Fantechi [4], induced by the Hessian of f . Since these symmetric ob-

struction theories do not necessarily glue as two-term complexes (see [20, Example 2.17]), a

d-critical scheme (X , s ) cannot be equipped with a global symmetric perfect obstruction the-

ory in general. However, the compatibility between d-critical charts is enough to ensure that

these locally defined obstruction theories give rise to a slightly weaker structure on X with

similar properties, called an almost perfect obstruction theory, as was shown in [21].

On the other hand, Pantev–Toën–Vaquié–Vezzosi [23] defined k -shifted symplectic struc-

tures on derived schemes and derived Artin stacks (we recall their definition in Section 1.1),

for every k ∈ Z. It is explained in [23, Section 3.2] that every −1-shifted symplectic structure

ω on a derived scheme X induces a symmetric perfect obstruction theory on the underlying

classical scheme X = t0(X ) ,→ X . Furthermore, as we recall in Theorem 1.3, there is a trunca-

tion functor

(0.1)−1-shifted symplectic derived Artin stacks

τ−→

d-critical Artin stacks

which takes (X ,ω) 7→ (X , sω), where X = t0(X ) ,→ X in the underlying classical Artin stack

of X and sω ∈H0(S0X) is a natural d-critical structure constructed out of ω. For a −1-shifted

symplectic derived scheme (X ,ω), the induced symmetric perfect obstruction theory on X =

t0(X ) ,→ X is isomorphic to the almost perfect obstruction theory arising from (X , sω).

The structures described so far are reproduced in Figure 1, inspired from a larger picture in

[20], where a dotted arrow means that the association only works locally. We will explain the

question mark ‘?’ appearing in the diagram in the next subsection.

truncation τ

?

moduli of sheaves

on a Calabi–Yau 3-fold

−1-shifted symplectic

derived schemes [23]

Joyce’s d-critical

schemes [20]

schemes with symmetric

perfect obstruction theory [4]

schemes with almost perfect

obstruction theory [21]

FIGURE 1. Landscape of the structures and their relations appearing in this

paper.

1Strictly speaking, a choice of orientation data is also needed, see [7] and [10] for the precise statements.

The d-critical structure on the Quot scheme of points of a Calabi–Yau 3-fold 3

0.2. Motivation. A classical example of−1-shifted symplectic derived scheme is that of a de-

rived critical locusRcrit( f ), for f ∈ O (V ) a regular function on a smooth scheme V [23, Corol-

lary 2.11]. The space Rcrit( f ) is defined as the derived fibre product of the zero section of

ΩV , carrying its canonical symplectic structure, with the section d f ∈ H0(V ,ΩV ). The −1-

shifted symplectic structure is denoted ω f in this case. We can view the classical scheme

U = crit( f ) = t0(Rcrit( f )) as a d-critical locus with d-critical structure

(0.2) sf = f + (d f )2 ∈H0(S0U )

determined by a single d-critical chart, and the diagonal dotted arrow in Figure 1 can in fact,

in this special case, be filled in by means of the critical symmetric obstruction theory

(0.3) Ef =TV

U

Hess( f )−−−−→ΩV

U

→LU ,

where LY is the truncated cotangent complex of a scheme Y . Moreover, in this case, the

functor τ in (0.1) sends (Rcrit( f ),ω f ) to (crit( f ), sf ).

A further example of a−1-shifted symplectic derived scheme is the derived moduli scheme

M X (ch)of simple coherent sheaves on a projective Calabi–Yau 3-fold X , with fixed Chern char-

acter ch ∈H∗(X ,Q), see [23]. In particular, the underlying classical scheme MX (ch) is naturally

a d-critical locus via the truncation functor (0.1). However, as Behrend pointed out in [2], it is

“an embarassment of the theory” that one cannot construct the d-critical structure on MX (ch)

directly, i.e. without passing through derived Algebraic Geometry. This is the meaning of the

question mark in Figure 1.

This “embarassment” was our main motivation for starting this project. This paper dis-

solves such embarassment in the following sense. On the Quot scheme of n points on A3,

namely the space

(0.4) Qr,n =QuotA3 (O⊕r , n ) =

[O ⊕r ։ E ]

dim E = 0, χ (E ) = n

,

there are in principle two d-critical structures:

(1) one arising from its critical structure crit( fr,n ) e→Qr,n [1] (and so a priori independent

of derived geometry),

(2) another arising as follows: the derived moduli stack Mn of 0-dimensional sheaves of

length n onA3 is−1-shifted symplectic [9], so by truncation one has a d-critical struc-

ture s dern on the underlying classical Artin stack Mn . Its pullback along the (smooth)

morphism Qr,n →Mn forgetting the surjection defines yet another d-critical struc-

ture on Qr,n .

We prove that the d-critical structures described in (1) and (2) agree, which shows that the

d-critical structure coming from derived geometry is ‘morally underived’, and is the simplest

possible.

We also make some progress in the projective case. More precisely, let F be a locally free

sheaf of rank r > 0 on a projective Calabi–Yau 3-fold X . There is a natural ‘derived’ d-critical

structure on the Quot scheme of points QuotX (F, n ). We devote Section 5 to showing that such

d-critical structure looks étale locally like the d-critical structure in (2) above, which in turn

agrees with the one induced by the unique d-critical chart on the local model QuotA3 (O ⊕r , n ).

0.3. Main results. We discuss in greater details our main results in the rest of this introduc-

tion.


0.3.1. The local case. The Quot scheme of points (0.4) parametrising isomorphism classes

of quotients O ⊕r ։ E , where E is a 0-dimensional sheaf of length n on A3, is proven in [1,

Theorem 2.6] to be a global critical locus (cf. Theorem 2.3). More precisely, there is a diagram

(0.5) ncQuotnr ⊃ crit( fr,n ) Qr,n

←

→ιr,n

∼

where fr,n is a regular function on the noncommutative Quot scheme ncQuotnr , a smooth (2n2+

r n )-dimensional variety which can be viewed as the moduli space of (isomorphism classes

of) (1, n )-dimensional stable r -framed representations (A, B , C , v1, . . . , vr )∈ EndC(Cn )3×(Cn )r

of the 3-loop quiver (cf. Figure 3). The function fr,n ∈ O (ncQuotnr ) is defined to be the trace of

the potential A[B , C ]. It defines a d-critical structure as in (0.2), namely

s critr,n = sfr,n

= fr,n + (d fr,n )2 ∈ H0

S

0crit( fr,n )

.

On the other hand, derived symplectic geometry endows the derived moduli stack Mn of

0-dimensional sheaves of length n on A3 with a −1-shifted symplectic structureωn (see [9]),

which can be truncated to produce a d-critical structure s dern = sωn

on the (underived) moduli

stack Mn = t0(Mn ). The morphism qr,n : Qr,n →Mn sending [O ⊕r ։ E ] 7→ [E ] is smooth

(cf. Section 1.4), so the pullback

s derr,n = q ∗r,n s der

n

defines a d-critical structure on Qr,n (cf. Definition 4.3).

The following is our first main result.

Theorem A. Fix r ≥ 1 and n ≥ 0. There is an identity

ι∗r,n s derr,n = s crit

r,n ∈H0S

0crit( fr,n )

.

0.3.2. The global case. Consider now the case where A3 is replaced by a smooth, projective

Calabi–Yau 3-fold X and O ⊕r by a locally free sheaf F of rank r on X . Let QF,n =QuotX (F, n )

be the Quot scheme parametrising quotients [F ։ E ]with E a 0-dimensional sheaf of length

n on X . The derived moduli stack MX (n ) of 0-dimensional sheaves of length n on X carries a

canonical −1-shifted symplectic structure by [23], which can be truncated to give a d-critical

structure sX ,n ∈ H0(S0MX (n )

). Its pullback along the forgetful morphism QF,n →MX (n ) is a

d-critical structure on the Quot scheme, denoted sF,n , see Equation (5.1).

The methods used in the proof of Theorem A also yield (after a bit of work) the following

result, which says that the d-critical scheme (QF,n , sF,n ) is locally modelled on the d-critical

scheme (Qr,n , s derr,n ).

Theorem B. Fix r ≥ 1 and n ≥ 0. Let X be a projective Calabi–Yau 3-fold, F a locally free sheafof rank r on X . There exists an analytic open cover ρλ : Tλ ,→ QF,nλ∈Λ, such that for anyindex λ we have a diagram

Tλ

Qr,n QF,n

←→

πλ ←-

→ρλ

with πλ étale, satisfying π∗λs derr,n =ρ

∗λsF,n ∈H0(S0

Tλ).


0.3.3. The two obstruction theories on Hilbn A3. Our third main result is a comparison be-

tween perfect obstruction theories on the Hilbert scheme of points Hilbn A3. If Vn = ncQuotn1

is the noncommutative Hilbert scheme, Diagram (0.5) becomes

(0.6) Vn ⊃ crit( f1,n ) Hilbn A3,←

→ι1,n

∼

and the Hessian construction (0.3) defines a symmetric obstruction theory

(0.7) Ef1,nLcrit( f1,n )

.←

→ϕcrit

On the other hand, viewing the Hilbert scheme as a parameter space for ideal sheaves of

colength n , one obtains the symmetric obstruction theory (more details are found in [24, 25])

Eder =Rπ∗RHom (I,I)0[2] LHilbn A3 ,←

→ϕder

where I ⊂ OA3×Hilbn A3 is the universal ideal sheaf, π: A3 ×Hilbn A3 → Hilbn A3 is the second

projection, and RHom (I,I)0 denotes the [−1]-shifted cone of the trace map RHom (I,I)→O .

The following result, established in Section 6, proves Conjecture 9.9 in [14].

Theorem C. The isomorphism ι1,n in (0.6) induces an isomorphism of perfect obstructiontheories

ι∗1,nEder Ef1,n

Lcrit( f1,n )

←

→∼

←

→ι∗1,nϕder

←

→ ϕcrit

Conventions. We work overC throughout. The ‘font’ used in this paper for schemes, derived

schemes, stacks and derived stacks will be X , X , X and X respectively. For a quasiprojective

variety X , we letMX (n ) denote the moduli stack of 0-dimensional coherent sheaves of length

n on X . If F is a coherent sheaf on X , we also set QF,n =QuotX (F, n ), where the right hand side

is Grothendieck’s Quot scheme, parametrising quotients F ։ E where E is a 0-dimensional

sheaf of length n . When (X , F ) = (A3,O ⊕r ), we set Mn =MA3 (n ) and Qr,n =QuotA3 (O ⊕r , n ).

Acknowledgments. We wish to thank Pierrick Bousseau, Ben Davison, Dominic Joyce, Dra-

gos Oprea, Sarah Scherotzke and Nicolò Sibilla for helpful discussions.

A.R. thanks SISSA for the excellent working conditions.

1. BACKGROUND MATERIAL

1.1. Shifted symplectic structures. Let cdga≤0C be the category of non-positively graded com-

mutative differential graded C-algebras. There is a spectrum functor Spec : cdga≤0C → dStC to

the category of derived stacks (see [33, Definition 2.2.2.14] or [31, Definition 4.2]). An object

of the form Spec A is called an affine derivedC-scheme. Such objects provide the Zariski local

charts for general derived C-schemes, see [31, Section 4.2]. An object X in dStC is called a de-

rived Artin stack if it is m-geometric (cf. [31, Definition 1.3.3.1]) for some m and its ‘classical

truncation’ t0(X ) is an Artin stack (and not a higher stack). A derived Artin stack X admits an

atlas, i.e. a smooth surjective morphism U →X from a derived scheme. Every derived Artin

stack X has a cotangent complex LX of finite cohomological amplitude in [−m , 1] and a dual

tangent complex TX . Both are objects in a suitable stable∞-category Lqcoh(X ) (see [31] or

[33] for its definition).


Shifted symplectic structures on derived Artin stacks were introduced by Pantev–Toën–

Vaquié–Vezzosi in [23]. The definition is given in the affine case first, and then generalised

by showing the local notion satisfies smooth descent. We recall the local definition: let us

set X = Spec A, so that Lqcoh(X )∼= D(dg-ModA). For all p ≥ 0 one can define the exterior

power complex (ΛpLX , d) ∈ Lqcoh(X ), where the differential d is induced by the differential

of the algebra A. For a fixed k ∈ Z, define a k -shifted p -form on X to be an element ω0 ∈

(ΛpLX )k such that dω0 = 0. To define the notion of closedness, consider the de Rham dif-

ferential ddR : ΛpLX → Λp+1LX . A k -shifted closed p -form is a sequence (ω0,ω1, . . .), with

ωi ∈ (Λp+iLX )k−i , such that dω0 = 0 and ddRω

i + dωi+1 = 0. When p = 2, any k -shifted 2-

form ω0 ∈ (Λ2LX )k induces a morphism ω0 : TX → LX [k ] in Lqcoh(X ), and we say that ω0 is

non-degenerate if this morphism is an isomorphism in Lqcoh(X ).

Definition 1.1 ([23, Definition 1.18]). A k -shifted closed 2-form ω = (ω0,ω1, . . .) is called a

k -shifted symplectic structure if ω0 is non-degenerate. We say that (X ,ω) is a k -shifted sym-

plectic (affine) derived scheme.

1.2. d-critical schemes and Artin stacks. Let X be a scheme over C. Joyce [20] proved the

existence of a canonical sheaf of C-vector spaces SX such that for every triple (R , V , i ), where

R ⊂ X is an open subscheme, V is a smooth scheme and i : R ,→ V is a closed immersion with

idealI , one has an exact sequence

0 SX

R

OV /I2

ΩV /I ·ΩV ,←

→

←

→

←

→d

where the last map is induced by the exterior derivative; see [20, Theorem 2.1] for the full list

of properties characterising SX . Joyce also proved the existence of a subsheaf S0X ⊂ SX and a

direct sum decomposition

SX =S0X ⊕CX ,

whereCX is the constant sheaf on X and, for any triple (R , V , i ) as above, one has

S0X

R= ker

SX

R

OV /I2 ORred

⊂ SX

R

.←- →

←

։

If V is a smooth scheme carrying a regular function f ∈ O (V )with critical locus R = crit( f )⊂

V , and f |Rred= 0, thenI = (d f )⊂OV and a natural element of H0(S0

X |R ) is the section f +(d f )2.

Definition 1.2 ([20, Definition 2.5]). A d-critical scheme is a pair (X , s ), where X is an ordinary

scheme and s is a section of S0X with the following property: for every point p ∈ X there is a

quadruple (R , V , f , i ), called a d-critical chart, where R ,→ X is an open neighbourhood of p ,

V is a smooth scheme, f ∈ O (V ) is a regular function such that f |Rred= 0, having critical locus

i : R ,→ V , and s |R = f + (d f )2 ∈H0(S0X |R ). The section s is called a d-critical structure on X .

By [20, Proposition 2.8], if g : X → Y is a smooth morphism of schemes and t ∈H0(S0Y ) is a

d-critical structure on Y , then g ∗t ∈H0(S0X ) is a d-critical structure on X .

For an Artin stack X , Joyce defined the sheaf S0X

by smooth descent [20, Corollary 2.52].

Recall that to give a sheafF onX one has to specify an étale sheafF (U , u ) for every smooth 1-

morphism u : U →X from a scheme, along with a series of natural compatibilities. Similarly,

to give a section s ∈H0(F ) is to give a collection of compatible sections s (U , u ) ∈H0(F (U , u ))

for every smooth 1-morphism u : U →X from a scheme.


Joyce defined a d-critical Artin stack (see [20, Definition 2.53]) to be a pair (X , s ), where X

is a classical Artin stack, s ∈ H0(S0X) is a section such that s (U , u ) defines a d-critical struc-

ture on U (being a section of S0X(U , u ) = S0

U ) according to Definition 1.2, for every smooth

1-morphism u : U →X .

If G is an algebraic group acting on a scheme Y , the sheaf S0Y is naturally G -equivariant,

so there is a well-defined subspace H0(S0Y )

G ⊂H0(S0Y ) of G -invariant sections. If X = [Y /G ],

then H0(S0X) =H0(S0

Y )G , and the d-critical structures on X are canonically identified with the

G -invariant d-critical structures on Y , see [20, Example 2.55]. We will make this identification

throughout without further mention.

Theorem 1.3 ([5, Theorem 3.18]). Let (X ,ω) be a −1-shifted symplectic derived Artin stack.Then the underlying classical Artin stack X = t0(X ) extends in a canonical way to a d-criticalArtin stack (X , sω). This defines a ‘truncation functor’ τ, as in (0.1), from the∞-category of−1-shifted symplectic derived Artin stacks to the 2-category of d-critical Artin stacks.

See also [8, Theorem 6.6] for the analogous result proved for schemes.

1.3. Symmetric obstruction theories. Let M be a C-scheme with full cotangent complex

L•M ∈D(−∞,0](QCohM ), and let LM ∈D[−1,0](QCohM ) denote its cutoff at −1. A perfect obstruc-

tion theory on M , as defined by Behrend–Fantechi [3], is a pair (E,φ), where E is a perfect

complex of perfect amplitude contained in [−1, 0], andφ : E→LM a morphism in the derived

category, such that h 0(φ) is an isomorphism and h−1(φ) is onto. A perfect obstruction theory

(E,φ) is called symmetric if there exists an isomorphism ϑ : E e→E∨[1] such that ϑ∨[1] = ϑ. See

[4] for background on symmetric obstruction theories.

If (M ,ω) is a−1-shifted symplectic derived scheme, with underlying classical scheme i : M ,→

M . Then E = i ∗LM → LM is a perfect obstruction theory, which is furthermore symmetric

thanks to the non-degenerate pairing ϑ = i ∗ω0. This association explains the left vertical ar-

row in Figure 1.

1.4. Smoothness of the forgetful map. Let X be a quasiprojective variety, n ≥ 0 an integer

and F ∈Coh(X ) a coherent sheaf on X . Set QF,n =QuotX (F, n ) and let π: QF,n ×X → X be the

projection. Forgetting the surjection defining the universal quotient

[π∗F ։Q] 7→ Q

defines a map qF,n : QF,n →MX (n ) to the moduli stack of length n coherent sheaves on X .

Such a map exists sinceQ ∈ Coh(QF,n × X ) is flat over QF,n by definition of the Quot scheme.

We call qF,n the forgetful morphism throughout.

We will need the following result.

Proposition 1.4. Let X be a reduced projective variety, F a locally free sheaf of rank r ≥ 1 onX . For every n ≥ 0, the forgetful morphism

qF,n : QF,n →MX (n )

is smooth of relative dimension r n .

Before proving the result, we recall a classical result by Grothendieck. According to [16,

Théorème 7.7.6] (but see also [22, Theorem 5.7]), if f : Y → B is a proper morphism to a locally


noetherian scheme B , and E is a coherent B -flat sheaf on Y , there exists a coherent sheafQE

on B along with functorial isomorphisms

η: f∗(E ⊗OBM ) HomOB

(QE ,M )←

→∼

for all quasicoherent sheavesM on B . The sheafQE is unique up to a unique isomorphism,

it behaves well with respect to pullback, and moreover it is locally free exactly when f is co-

homologically flat in dimension 0 [16, Proposition 7.8.4]. For instance, any proper flat mor-

phism with geometrically reduced fibres is cohomologically flat in dimension 0 [16, Proposi-

tion 7.8.6].

Proof of Proposition 1.4. Let XF,n be the stack of pairs (E ,α), where E is a 0-dimensional

sheaf of length n and α is an OX -linear homomorphism α: F → E . Then we have an open

immersion QF,n ,→XF,n and the morphism qF,n extends to a morphism

πF,n : XF,n →MX (n ).

Let B be a scheme. By standard methods, we may reduce to the case where B is locally noe-

therian. Given a map B →MX (n ), let us consider the fibre products

P QF,n

B MX (n )

←

→

←→ ←→ qF,n

←

→

and

V XF,n

B MX (n )

←

→←→ ←→ πF,n

←

→

where B →MX (n ) corresponds to a B -flat family of 0-dimensional sheaves E ∈ Coh(X × B ).

The associated sheafQE ∈Coh(B ) is locally free of rank r n , since the projection f : X ×B → B

is cohomologically flat in dimension 0 since it is proper with reduced fibres. Moreover, the

projection V → B is canonically isomorphic to the structure morphism

Spec SymOBQE → B ,

because of Grothendieck’s theorem [16, Cor. 7.7.8, Rem. 7.7.9], which says the following: Let

f : Y → B be a projective morphism, F and E two coherent sheaves on Y . Consider the

functor SchopB → Sets sending a B -scheme T → B to the set of morphism HomYT

(FT ,ET ),

whereFT and ET are the pullbacks ofF and E along the projection YT = Y ×B T → Y . Then,

if E is flat over B , the above functor is represented by a linear scheme Spec SymOBH → B ,

whereH is a coherent sheaf on B . However, in our case we have Y = X ×B , f = p2 the second

projection and F = p ∗1 F , and the functor described above is precisely the functor of points

of V , thus we must haveH =QE . More details can be found in the proof of Grothendieck’s

result found in [22, Theorem 5.8].

Therefore, since P ,→ V is open and V → B is an affine bundle of rank r n , the projection

P → B is smooth of relative dimension r n .

Remark 1.5. Let X ,→ X be an open subscheme, where X is a reduced projective variety as

in Proposition 1.4, and let F = F |X be the restriction of a locally free sheaf F over X . Then

qF ,n : QuotX (F, n )→MX (n )

is again smooth, being the pullback of the smooth morphism qF,n : QF,n →MX (n ) along the

open immersion MX (n ) ,→MX (n ). This applies for instance to (X , F, X ) = (P3,O ⊕r ,A3).

Thus we get the next corollary.


Corollary 1.6. The forgetful map qr,n : QuotA3 (O ⊕r , n )→Mn is smooth of relative dimensionr n .

Note that, for fixed [E ] ∈Mn , the choice of a surjection O ⊕r ։ E is nothing but the datum

of r (general enough) sections σ1, . . . ,σr ∈Hom(O , E ) =H0(E ) =Cn . So the fibre of qr,n over

[E ] is an open subset of Cr n .

2. PROOF OF THEOREM A

In this section we prove Theorem A (see Theorem 2.6) granting the (fundamental) auxiliary

result Theorem 2.2, which will be proved in Theorem 3.10.

The moduli stack Mn of 0-dimensional coherent sheaves on length n on A3 can be seen

as a stack of representations of the Jacobi algebra of a quiver with potential. Indeed, consider

the 3-loop quiver L3 (Figure 2), equipped with the potential W = A[B , C ].

1

A

B

C

FIGURE 2. The 3-loop quiver L3.

Notation 2.1. Fix a quiver Q = (Q0,Q1, s , t ), where Q0 is the vertex set, Q1 is the edge set, s and

t are the source and target maps Q1→Q0 respectively; for a dimension vector d = (d i )i ∈NQ0 ,

we let Repd (Q ) denote the space of d -dimensional representations of Q , namely the affine

space∏

a∈Q1HomC(C

d s (a ) ,Cd t (a ) ). The group GLd =∏

i∈Q0GLd i

acts on Repd (Q ) by conju-

gation; the moduli stack of d -dimensional representations is the quotient stack Md (Q ) =

[Repd (Q )/GLd ].

The critical locus of the GLn -invariant regular function

(2.1) fn = (Tr W )n : Repn (L3)→C, (A, B , C ) 7→ Tr A[B , C ],

defines a closed GLn -invariant subscheme Un of the 3n2-dimensional affine space Repn (L3)∼=

EndC(Cn )3 parametrising triples of pairwise commuting endomorphisms. There is an isomor-

phism of Artin stacks

(2.2) ιn : [Un/GLn ] Mn

←

→∼

defined on closed points by sending the GLn -orbit [A, B , C ] of a triple (A, B , C ) ∈ Un to the

point [E ] represented by the direct sum E = Op1⊕ · · · ⊕ Opn

, where the points pi are not nec-

essarily distinct, and are determined by the diagonal entries of the matrices A (for the x -

coordinate), B (for the y -coordinate) and C (for the z -coordinate). Indeed, the matrices can

be simultaneously put in upper triangular form precisely because they pairwise commute,

and we are working modulo GLn . The source of the isomorphism ιn is equipped with the

algebraic d-critical structure

s critn = fn + (d fn )

2 ∈H0S

0[Un/GLn ]

=H0

S

0Un

GLn.


On the other hand, Brav and Dyckerhoff [9] proved that there is a−1-shifted symplectic struc-

tureωn on the derived Artin stack Mn of 0-dimensional coherent sheaves of length n onA3

(see Subsection 3.2 for details). We denote by

(2.3) (Mn , s dern ) = τ(Mn ,ωn )

its truncation, defined through the ‘truncation functor’ τ recalled in Theorem 1.3.

The following result, granted for now, will be proved in Section 3.4.

Theorem 2.2. The isomorphism (2.2) induces an identity

ι∗n s dern = s crit

n

of algebraic d-critical structures.

Now, for a fixed integer r ≥ 1, let us consider the Quot scheme Qr,n =QuotA3 (O ⊕r , n ). As we

now recall, this Quot scheme is a global critical locus, or, in other words, it admits a d-critical

structure consisting of a single d-critical chart. This result in the case r = 1 is due to Szendroi

[28, Theorem 1.3.1]. The ‘r -framed 3-loop quiver’ eL3 (Figure 3) plays a crucial role.

∞ 1

A

B

C

...

FIGURE 3. The r -framed 3-loop quiver eL3.

Theorem 2.3 ([1, Theorem 2.6]). Fix a vector θ = (θ1,θ2) ∈R2, with θ1 > θ2. Let

ncQuotnr =Repθ−st

(1,n )(eL3)/GLn

be the moduli space of θ -stable representations of the r -framed 3-loop quiver eL3, and con-sider the regular function fr,n : ncQuotn

r → A1 induced by the potential W = A[B , C ]. Then

there is an isomorphism

ιr,n : crit( fr,n ) Qr,n .←

→∼

The main result of this paper (Theorem A, proved in Theorem 2.6 below) states that the

d-critical structure

s critr,n = fr,n + (d fr,n )

2 ∈H0S

0crit( fr,n )

determined by the function fr,n in Theorem 2.3 agrees, up to the isomorphism ιr,n , with the

pullback of s dern along the smooth forgetful morphism qr,n .

Remark 2.4. According to our conventions, in a dimension vector (or a stability condition) on

a framed quiver∞→Q the first entry always refers to the framing vertex∞. As explained in

[1, 12], stability of a representation (A, B , C , v1, . . . , vr ) ∈ Rep(1,n )(eL3)with respect toθ translates

into the condition that the framing vectors v1, . . . , vr ∈ HomC(V∞, V1) = Cn generate the un-

derlying (unframed) representation (A, B , C ) ∈ Repn (L3). Since GLn acts freely on Repθ−st(1,n )(

eL3),

this exhibits ncQuotnr as a smooth quasiprojective variety of dimension 2n2 + r n .


We have a cartesian diagram

(2.4)

crit( fr,n ) Qr,n

[Un/GLn ] Mn

←

→eqr,n

←

→ιr,n

∼

←

→ qr,n

←

→ιn∼

where ιr,n is the isomorphism of Theorem 2.3 and ιn is the isomorphism (2.2).

Proposition 2.5. There is an identity of d-critical structures

eq ∗r,n s critn = s crit

r,n ∈H0S

0crit( fr,n )

,

where s critr,n = fr,n + (d fr,n )

2 is the d-critical structure defined by the function fr,n .

Proof. Forgetting the framing data yields a smooth morphism

pr,n : ncQuotnr →Mn (L3) = [Repn (L3)/GLn ]

fitting in a cartesian diagram

crit( fr,n ) ncQuotnr

[Un/GLn ] Mn (L3)

←

→eqr,n

←- →←

→ pr,n

←- →

where the horizontal arrows are closed immersions. The function fn introduced in (2.1) de-

scends to a function Mn (L3)→A1, still denoted fn , with critical locus [Un/GLn ]. The conclu-

sion then follows by observing that p ∗r,n fn = fr,n , which is true because fr,n does not interact

with the framing data.

We can now complete the proof of Theorem A.

Theorem 2.6. Let s derr,n ∈ H0(S0

Qr,n) be the pullback of the d-critical structure s der

n (defined in(2.3) by truncating ωn ) along qr,n . Then the isomorphism ιr,n of Theorem 2.3 induces anidentity


r,n ∈H0S

0crit( fr,n )

.

Proof. We have

ι∗r,n s derr,n = ι

∗r,n q ∗r,n s der

n by definition

= eq ∗r,n ι∗n s der

n by Diagram (2.4)

= eq ∗r,n s critn by Theorem 2.2

= s critr,n by Proposition 2.5,

which concludes the proof.

3. THE D-CRITICAL STRUCTURE(S) ON Mn

In this section we compare the two d-critical structures

s critn ∈H0

S

0[Un/GLn ]

, s der

n ∈H0S

0Mn


on the isomorphic spaces [Un/GLn ] and Mn , introduced in Section 2. These d-critical struc-

tures come from quiver representations and symplectic derived algebraic geometry respec-

tively. Our goal (achieved in Theorem 3.10) is to prove Theorem 2.2.

To do so, we first analyse and give explicit local d-critical charts for the two d-critical struc-

tures. For s critn , we use the geometry of the quotient stack [Un/GLn ] and Luna’s étale slice

theorem. For s dern , we take advantage of the derived deformation theory of the −1-shifted

symplectic stack Mn to obtain explicit formal charts in Darboux form (cf. [5]). Finally, we

argue that one can pass from formal neighbourhoods to honest smooth neighbourhoods to

obtain the equality of the d-critical structures.

Notation 3.1. Throughout we shall use the notation Mata ,b (R ) to denote the space of matrices

with a rows and b columns with entries in a ring R . If R =C, we simply write Mata ,b .

3.1. The d-critical structure s critn coming from quiver representations. Let [E ] ∈Mn be a

closed point. Then the corresponding sheaf E must be polystable, i.e. of the form

(3.1) E =k⊕

i=1

Cai ⊗Opi

where, for i = 1, . . . , k , the points pi = (αi ,βi ,γi ) ∈A3 are pairwise distinct and ai are positive

integers such that a1 + · · ·+ak = n .

We fix some notation for convenience. Let Yn = Repn (L3) = EndC(Cn )3 be the vector space

of triples of n ×n matrices, and let fn : Yn → C be the regular function (2.1). Denote by a =

(a1, . . . , ak ) the k -tuple of positive integers determined by (3.1). Form the product

Ya =

k∏

i=1

Yai.

Then there is a closed embedding Φa : Ya ,→ Yn by block diagonal matrices with square diag-

onal blocks of sizes a1, . . . , ak . The reductive algebraic group GLa =∏k

i=1 GLaiacts on Ya by

componentwise conjugation and Φa is equivariant with respect to the inclusion GLa ⊂ GLn

by block diagonal matrices of the same kind.

On the space Ya we have the GLa -invariant potential

(3.2) ga = fa1⊕ · · ·⊕ fak

: Ya →A1, (Ai , Bi , Ci )i 7→

k∑

i=1

Tr Ai [Bi , Ci ],

where (Ai , Bi , Ci ) ∈ Yaifor i = 1, . . . , k . We have the obvious relation fn Φa = ga . If we set

Un = crit( fn )⊂ Yn , Ua = crit(ga )⊂ Ya ,

the restriction Φa : Ua ,→Un is still equivariant with respect to the inclusion GLa ⊂ GLn , and

so it induces a morphism of algebraic stacks

(3.3) ψa : [Ua /GLa ]→ [Un/GLn ].

Note that the identification crit(ga ) =∏

i crit( fai) =

∏i Uai

combined with the product of the

isomorphisms ιaias in (2.2) induces a canonical identification

[Ua/GLa ] =

k∏

i=1

[Uai/GLai

]

k∏

i=1

Mai=Ma .←

→∼


We can identify the mapψa with the direct sum map

ψa : Ma →Mn ,

denoted the same way, taking a B -valued point of Ma , i.e. a k -tuple (E1, . . . ,Ek ) of B -flat fami-

lies of 0-dimensional sheavesEi ∈Coh(B×A3), to their direct sum⊕

1≤i≤k Ei . (The direct sum

of flat sheaves if flat by [27, Tag 05NC]).

Lemma 3.2. Let Xa ⊂Ma be the open substack parametrising k -tuples of sheaves with pair-wise disjoint support. Then the morphism

ψa

Xa

: Xa →Mn

is étale.

Proof. According to [27, Tag 0CIK], to show that ψ ..= ψa |Xais étale it is enough to find an

algebraic space W , a faithfully flat morphism ρ : W →Mn locally of finite presentation and

an étale morphism T → W ×ρ,Mn ,ψ Xa such that T →W is étale. We start by picking W =

QuotA3 (O ⊕n , n ) along with the smooth (cf. Corollary 1.6) morphism

ρ = qn ,n : QuotA3 (O⊕n , n )→Mn

sending, for any test scheme B , a B -flat quotient O ⊕nB×A3 ։ T to the object T ∈Mn (B ). Note

that ρ is surjective in the sense of [27, Tag 04ZR]. Indeed, for any closed point [E ], we have a

direct sum decomposition as in (3.1), thus the sheaf E receives a surjection from O ⊕n , which

is nothing but the direct sum of the surjections O ⊕ai ։Cai ⊗Opi— their direct sum is again

surjective because pi 6= pj for all i 6= j .

Next, we form the cartesian diagram

T Xa

QuotA3 (O ⊕n , n ) Mn

←

→µ

←

→

←

→ ψ

←

→ρ

and we pick the identity T = QuotA3 (O ⊕n , n )×ρ,Mn ,ψXa as an étale map. We need to show

that µ is étale. But this follows from [1, Proposition A.3], after observing that T is the open

subscheme ofk∏

i=1

QuotA3 (O⊕n , ai )

parametrising k -tuples of quotients with pairwise disjoint support, and µ is nothing but the

map taking a tuple of surjections to their direct sum.

Let E be a polystable sheaf supported on points pi = (αi ,βi ,γi ) for 1≤ i ≤ k , as in (3.1). The

point [E ] ∈Mn corresponds under ιn to the orbit of the triple of matrices yE =Φa (vE ) ∈Un ⊂

Yn , where vE = (α,β ,γ) ∈Ua ⊂ Ya is given by

(3.4) α= (α1 Ida1, . . . ,αk Idak

), β = (β1 Ida1, . . . ,βk Idak

), γ= (γ1 Ida1, . . . ,γk Idak

).

We let v E denote the image of vE along the smooth atlas Ua → [Ua /GLa ], and similarly for

y E ∈ [Un/GLn ]. The morphismψa maps v E ∈ [Ua /GLa ] to y E ∈ [Un/GLn ]. For convenience,

we will identify α,β ,γ with their images in Yn under Φa , considering them as n × n block

diagonal matrices and more generally we identify elements of Ya with their image in Yn under

Φa .

https://stacks.math.columbia.edu/tag/05NC

https://stacks.math.columbia.edu/tag/0CIK

https://stacks.math.columbia.edu/tag/04ZR


Lemma 3.3. There is an open GLa -invariant neighbourhood vE ∈ V ⊂ Ya such that, if U =

crit(ga |V ), the morphism ψE..=ψa |[U /GLa ]

is étale. In particularψa is étale at v E . Moreover,ifφE denotes the composition

φE : U [U /GLa ] [Ua /GLa ] [Un/GLn ],

←

→ ←- →

←

→ψa

then we have an identity of d-critical structures

φ∗E s critn = ga |V + (dga |V )

2 ∈H0(S0U )

GLa ⊂H0(S0U ).

Proof. The first statement follows from Lemma 3.2, up to replacing the open substack [U /GLa ] ,→

[Ua /GLa ]with its intersection with Xa . We now compare fn and ga .

For a complex matrix A ∈ Matn ,n (cf. Notation 3.1), we consider it as a block matrix with

diagonal blocks A11, . . . , Ak k of sizes a1 × a1, . . . , ak × ak and off-diagonal blocks Ai j of sizes

ai ×a j .

Let Yn = Ya ⊕ Y ⊥n be the direct sum decomposition where Y ⊥n is the subspace of triples of

matrices (A, B , C )whose diagonal blocks are zero. This decomposition is GLa -invariant, since

the conjugation action of GLa is given on blocks by the formula

(h · (A, B , C ))i j =hi i Ai j h−1

j j , hi i Bi j h−1j j , hi i Ci j h−1

j j

.

The derivative gln → TyEYn of the GLn -action on Yn at the point yE = (α,β ,γ) is given by the

map

σ : gln =Matn ,n −→ TyEYn ≃ Yn

X 7→[X ,α], [X ,β ], [X ,γ]

Notice that the i j -th blocks of the triple of matrices σ(X ) are given by the triple of ai × a j

matrices

(3.5) σ(X )i j =−(αi −α j )X i j , (βi −β j )X i j , (γi −γ j )X i j

.

In particular, im(σ) is a GLa -invariant subspace of Y ⊥n .

Define a subspace im(σ)⊥ ⊂ Yn by the condition that (X , Y , Z ) ∈ im(σ)⊥ if and only if for all

indices i 6= j we have

(3.6) (αi −α j )X i j + (βi −β j )Yi j + (γi −γ j )Zi j = 0.

We thus have a GLa -invariant direct sum decomposition

(3.7) TyEYn ≃ Yn = Ya ⊕ im(σ)⊕ Y σ

n

where Ya ⊕Y σn = im(σ)⊥ and Y σ

n is the GLa -invariant complement of Ya in im(σ)⊥ consisting

of the matrices that have zero diagonal blocks. Since im(σ) = TyE(GLn ·yE ), by Luna’s étale slice

theorem there exists a sufficiently small GLa -invariant open neighbourhood S of yE ∈ Ya⊕Y σn

which is an étale slice for the GLn -action on Yn at yE . Therefore, T =Un ×YnS is an étale slice

for the GLn -action on Un at yE . In particular, T is cut out by the equations d fn |S = 0 in S and

thus T = crit( fn |S )⊂ S .

Since fn |S = Tr A[B , C ]|S , it is easy to check that the derivatives of fn with respect to the

coordinates of Y σn vanish on Ya . Moreover, the Hessian H of fn at the point yE ∈ Yn is given

by the quadratic form whose value at (X , Y , Z ) ∈ TyEYn ≃ Yn is

(3.8) H (X , Y , Z ) = TrX [β , Z ]+X [Y ,γ]+α[Y , Z ]

.


By direct computation, the form H is non-degenerate on Y σn . To see this, working in terms

of blocks, we have

H (X , Y , Z ) =∑

i 6= j

Tr(βi −β j )Zi j X j i − (γi −γ j )Yi j X j i − (αi −α j )Zi j Yj i

.

For simplicity, we write each pair of summands corresponding to the indices i 6= j sugges-

tively in the form

(3.9) β Tr(Z1X2)−γTr(Y1X2)−αTr(Z1Y2)−β Tr(Z2X1) +γTr(Y2X1) +αTr(Z2Y1)

where X1, Y1, Z1 (resp. X2, Y2, Z2) stand for X i j , Yi j , Zi j (resp. X j i , Yj i , Z j i ) and α,β ,γ stand for

αi −α j ,βi −β j ,γi −γ j respectively. Then (3.6) takes the equivalent form

αX1 +βY1+γZ1 = 0

αX2 +βY2+γZ2 = 0.

At least one of α,β ,γ is non-zero, so without loss of generality we assume α 6= 0, as the

computation is analogous in the other two cases. Solving for X1 and X2, substituting into

(3.9) and simplifying, gives the expression

1

α

|α|2 + |β |2+ |γ|2

Tr (Y1Z2− Y2Z1) .

This is non-degenerate and therefore H must be non-degenerate.

Since fn |Ya= ga , the non-degeneracy of H on Y σ

n implies that, after possibly replacing S

by a GLa -invariant open neighbourhood of yE in S , we may assume that the ideal I = (d fn |S )

coincides with the ideal (dga , K ), where K is the ideal of Ya in S generated by the coordinates

of Y σn .

Thus, letting U =Ua ×S Ya, we see that U is an open neighbourhood of vE in Ua and the

étale map

κ=ψa

[U /GLa ]

: [U /GLa ]−→ [Ua /GLa ]−→ [Un/GLn ]

satisfies by construction κ∗s crit

n

U= fn

S+ I 2 ∈H0(S0

U )GLa .

Let V = Ya ×S Y σn . By definition, V is a GLa -invariant open neighbourhood of yE in Ya . We

have the embedding Ψ : V → S such that Ψ∗ fn |S = ga |V and U = crit(ga |V ). Write J for the

ideal (dga |V ). Then, by the definition of the sheaf S0U , the commutative diagram

0 S0U OS/I 2

ΩS/I ·ΩS

0 S0U OV / J 2

ΩV / J ·ΩV

←

→

⇐⇐

←

→

←→ Ψ∗

←

→d

←→ Ψ∗

←

→

←

→

←

→d

implies that fn |S + I 2 = ga |V + J 2 ∈H0(S0U )

GLa , which is exactly what we want.

3.2. The derived stack Mn of 0-dimensional coherent sheaves of length n onA3. Let Mn

be the derived moduli stack of 0-dimensional coherent sheaves of length n on A3. Its clas-

sical truncation is the stack Mn . We now recall the definition of Mn and give an explicit

description as a derived quotient stack for the convenience of the reader.

Let A ∈ cdga≤0C . By [32] or [9, Example 2.7], Mn assigns to A the mapping space

(3.10) MapdgCat

C[x , y , z ], Projn (A)

.


Here C[x , y , z ] is viewed as the dg-category with one object and morphisms concentrated in

degree 0 given by C[x , y , z ], whereas Projn (A) is the dg-category whose objects are rank n

projective A-modules and the morphism complexes are given by Hom•(E , F ) for two objects

E , F .

Since projective modules are locally trivial and, as differential graded algebras, we have

Hom•(An , An ) = gln ⊗C A, we can further simplify (3.10) (up to possible shrinking) into

(3.11) MapdgaC

C[x , y , z ], gln ⊗C A

,

up to conjugation by GLn .

To compute this mapping space with respect to the model structure of the category dgaC,

we need a resolution of the algebra C[x , y , z ] by a semi-free dg-algebra. This is provided by

the following proposition.

Proposition 3.4. LetQ3 be the semi-free dg-algebra with generators x , y , z in degree 0, x ∗, y ∗, z ∗

in degree −1 and w in degree −2 and differential δ satisfying

δx =δy =δz = 0

δx ∗ = [y , z ], δy ∗ = [z , x ], δz ∗ = [x , y ]

δw = [x , x ∗]+ [y , y ∗]+ [z , z ∗].

(3.12)

Then the natural morphism Q3→C[x , y , z ] is a quasi-isomorphism.

Proof. The claim follows from the fact that theC-algebraC ⟨x , y , z ⟩ generated overC freely by

three variables x , y , z is Calabi–Yau of dimension 3 and [15, Theorem 5.3.1].

It follows from the proposition that the mapping space (3.11) is computed by

HomdgaC (Q3, gln ⊗C A)

which, using the results of [6], is naturally isomorphic to

Homcdga≤0C((Q3)n , A) .

Here (Q3)n is the commutative dg-algebra with generators corresponding to the matrix entries

of the n × n matrices X , Y , Z in degree 0, X ∗, Y ∗, Z ∗ in degree −1 and W in degree −2 with

differential determined by (3.12). In particular we have sets of generators given by the matrix

entries of

Repn (L3)≃ gl⊕3n , Repn (L3)≃ gl

⊕3n , gln

in degrees 0, −1 and −2 respectively.

Quotienting by the conjugation action of GLn , we obtain the following proposition.

Proposition 3.5. There exists an isomorphism of derived Artin stacks

(3.13) ιn :Spec (Q3)n

GLn

Mn .←

→∼

3.3. The d-critical structure s dern coming from derived symplectic geometry. Recall that Equa-

tion (2.3) defines s dern to be the truncation of the −1-shifted symplectic structure ωn on Mn

constructed by Brav and Dyckerhoff [9].

Let [E ] ∈Mn be a closed point, so that the sheaf E is polystable as in (3.1). We now pro-

ceed to find formally smooth d-critical charts for (Mn , s dern ) around such points [E ] using the


derived deformation theory of Mn , following [30]. In order to do this, we first need some

preparations.

Let p = (a0, b0, c0) ∈A3 be a point in coordinates x0, y0, z0 for A3. We denote the functions

x0−a0, y0− b0, z0− c0 by x , y , z respectively. For convenience, we also write O in place of the

ring OA3 =C[x0, y0, z0] in what follows.

We have the Koszul resolution Q •p →Op , given by the complex

(3.14) Q •p =Q−3→Q−2→Q−1→Q 0

=O

A−→O ⊕3 B

−→O ⊕3 C−→O

,

where the morphisms A, B , C are multiplications by the matrices (we are very slightly abusing

notation here)

A =

x

y

z

, B =

0 −z y

z 0 −x

−y x 0

, C =x y z

.

Let

(3.15) gp =Hom(Q •p ,Q •p ) =RHom(Op ,Op ).

By composing morphisms in each degree, gp has the structure of an (infinite-dimensional)

dg-algebra (gp , ·,δ) overC. Moreover, let

(3.16) gminp =

3⊕

i=0

Hi (gp ) =

3⊕

i=0

Exti (Op ,Op ).

Using the Yoneda product

m2 : Exti (Op ,Op )⊗Ext j (Op ,Op )−→ Exti+ j (Op ,Op ),

one can endow gminp with the structure of a graded commutative algebra over C and thus of

a dg-algebra (gminp , m2, 0) with zero differential. As a graded algebra, it is clear that gmin

p is

isomorphic to the exterior algebra Λ•C3.

Lemma 3.6. There exists a morphism Ip : (gminp , m2, 0) → (gp , ·,δ) of dg-algebras, which in-

duces the identity map on homology. In particular, the dg-algebra gp is formal as an A∞-algebra.

Proof. We construct the morphism Ip explicitly. For brevity, we identify HomO (O⊕s ,O ⊕t )with

t × s matrices Matt ,s (O ) with entries in O .

The graded pieces of gp in degrees 0, . . . , 4 are

g0p =

⊕

i

Hom(Q i ,Q i ) =O ⊕Mat3,3(O )⊕Mat3,3(O )⊕O

g1p =

⊕

i

Hom(Q i ,Q i+1) =Mat3,1(O )⊕Mat3,3(O )⊕Mat1,3(O )

g2p =

⊕

i

Hom(Q i ,Q i+2) =Mat3,1(O )⊕Mat1,3(O )

g3p =

⊕

i

Hom(Q i ,Q i+3) =O

g4p =

⊕

i

Hom(Q i ,Q i+4) = 0.


Define elements b1∈ g0p and bx , by , bz ∈ g1

p by

b1= (1, Id3, Id3, 1)

bx =

1

0

0

,

0 0 0

0 0 −1

0 1 0

,1 0 0

by =

0

1

0

,

0 0 1

0 0 0

−1 0 0

,0 1 0

bz =

0

0

1

,

0 −1 0

1 0 0

0 0 0

,0 0 1

.

We may verify directly that these satisfy the relations

b1 · bx = bx , b1 · by = by , b1 · bz = bzbx · bx = 0, by · by = 0, bz · bz = 0,

as well as

bx · by =−by · bx =

0

0

1

,1 0 0

by · bz =−bz · by =

1

0

0

,1 0 0

bz · bx =−bx · bz =

0

1

0

,0 1 0

bx · by · bz = 2.

So the associative algebra R generated byb1, bx , by , bz is a sub-algebra ofgp isomorphic toΛ•C3.

Moreover, all of the elements of R are cocycles for the differential δ of gp and their images in

homology give bases for the homology groups Hi (gp ) = Exti (Op ,Op ). By the definition of the

Yoneda product m2, it is immediate that R is isomorphic to gminp , giving a morphism of dg-

algebras Ip : gminp → gp which, by construction, induces the identity on homology, as desired.

Remark 3.7. The element bx ∈ g1p is the evaluation of the triple (A, B , C ) at the point (1, 0, 0) ∈

C3. Analogous statements hold for by , bz and all products of bx , by , bz . This motivates their defi-

nition.

Let now E be a polystable sheaf as in (3.1) and write Q •E =⊕k

i=1Cai ⊗Q •pi

. We then define

gE =Hom(Q •E ,Q •E ) =k⊕

i , j=1

Hom(Q •pi,Q •pj

)⊗Mata j ,ai=RHom(E , E )


and

gssE =

k⊕

i=1

gpi⊗Matai ,ai

gminE =

k⊕

i=1

gminpi⊗Matai ,ai

.(3.17)

Then gE is a dg-algebra with differential δ induced from that of each summand gpiand gss

E ⊂

gE is a dg-subalgebra. In the same fashion gminE is a commutative graded algebra. Since for

any i 6= j and any ℓ we have Extℓ(Opi,Opj) = 0, the inclusion gss

E ⊂ gE is a quasi-isomorphism

and gminE is isomorphic as a dg-algebra to (Ext∗(E , E ), m2, 0).

Corollary 3.8. Let E be a polystable sheaf as in (3.1). Then there exists a morphism

IE : (gminE , m2, 0)→ (gE , ·,δ)

of dg-algebras, which induces the identity map on homology. Thus, the dg-algebra gE is for-

mal as an A∞-algebra.

Proof. The map IE is given by the composition of the morphism⊕k

i=1 Ipi⊗ idMatai ,ai

and the

inclusion gssE ⊂ gE .

As in [29, Appendix A], let S = bS (gmin,>0E [1])∨ be the dual of the abelianisation of the Bar

construction of the dg-algebra gmin,>0E , which is the Chevalley–Eilenberg dg-algebra of the dg-

Lie algebra associated to the commutative dg-algebra gminE , with Lie bracket on each gmin

pi⊗

Matai ,aidetermined by

(bx∨i ⊗Ai ), (by ∨i ⊗Bi )

= bz∨i ⊗ [Ai , Bi ]

(by ∨i ⊗Bi ), (bz∨i ⊗Ci )

= bx∨i ⊗ [Bi , Ci ]

(bz∨i ⊗Ci ), (bx∨i ⊗Ai )

= by ∨i ⊗ [Ci , Ai ]

(3.18)

where each gminpi

∼= Λ•C3 using the given basis bxi , byi , bzi . Here we have used the fact that the

Lie bracket in the dg-Lie algebra structure of the derived tangent complex gE [1] of Mn at [E ]

is given by commutator of matrices (see [9, Proposition 3.3] or [18, Proposition 2.4.4]).

Thus, the definition (3.17) implies that S is generated by subalgebras Si , where each Si is a

negatively graded commutative dg-algebra and the differential ε in degree −1 is determined

by the morphisms

Λ2(C3)∨⊗Mat∨ai ,ai

εi−→ Sym•

(C3)∨⊗Mat∨ai ,ai

(3.19)

satisfying, due to the Lie bracket relations (3.18),

(bx∨i ⊗Ai )∧ (by ∨i ⊗Bi ) 7→ bz∨i ⊗ [Ai , Bi ]

(by ∨i ⊗Bi )∧ (bz∨i ⊗Ci ) 7→ bx∨i ⊗ [Bi , Ci ]

(bz∨i ⊗Ci )∧ (bx∨i ⊗Ai ) 7→ by ∨i ⊗ [Ci , Ai ].

(3.20)

Let ÒYaibe the formal completion of Yai

= EndC(Cai )3 at the origin and let bfaibe the formal

function

bfai: ÒYai

→C, bfai(Ai , Bi , Ci ) = Tr Ai [Bi , Ci ].


Similarly, let

bga = bfa1⊕ · · ·⊕ bfak

: ÒYa =

k∏

i=1

ÒYai→C(3.21)

be the function bga (v1, . . . , vk ) = bfa1(v1) + · · ·+ bfak

(vk ).

It follows immediately by (3.19) that

(3.22) Spec H0(S) =

k∏

i=1

crit( bfai) = crit

k⊕

i=1

bfai

= crit(bga )⊂ ÒYa.

Then by [29, Appendix A], the morphism

Spec S→Mn

gives a formal atlas for Mn at the point [E ]which moreover is in Darboux form with respect

to the −1-shifted symplectic structureωn . In particular, the truncation

Spec H0 (S) = crit(bga )→Mn

gives a hull for Mn at [E ], which is a formal smooth d-critical chart with respect to the d-

critical structure s dern . We have established the following lemma.

Lemma 3.9. Let [E ] ∈Mn be a polystable sheaf of the form (3.1), and set ÒU = crit(bga ), wherebga : ÒYa →C is given by (3.21). Then there exists a formally smooth morphism

(3.23) ÒφE : ÒU →Mn ,

mapping 0 ∈ ÒU to [E ] ∈Mn , inducing an isomorphism at the level of tangent spaces and an

identity

(ÒφE )∗s der

n = bga + (dbga )2 ∈H0(S0

ÒU ).

3.4. The two d-critical structures are equal. We now prove Theorem 2.2, showing that the

two d-critical structures studied in this section actually coincide. Our argument follows closely

the reasoning outlined in [29, Appendix A].

Theorem 3.10. The isomorphism ιn : [Un/GLn ] e→Mn from (2.2) induces an identity

ι∗n s dern = s crit

n

of algebraic d-critical structures.

Proof. Let [E ] ∈Mn be a polystable sheaf, given by the expression (3.1). Applying Lemma 3.3,

we have an étale morphism ψE : [U /GLa ]→ [Un/GLn ] such that the pullback under the in-

duced morphism φE : U → [U /GLa ]→ [Un/GLn ] satisfies

(3.24) φ∗E s critn = ga |V + (dga |V )

2 ∈H0(S0U )

GLa ⊂H0(S0U ),

where V ⊂ Ya is an open neighbourhood of the point vE ∈ Ya corresponding to [E ] and ga is

given by (3.2). By Lemma 3.9, we have a formal atlas

ÒφE : ÒU →Mn , 0 7→ [E ],

satisfying

(3.25) Òφ∗E (sdern ) = bga + (dbga )

2 ∈H0(S0ÒU ).


Let jm : UE ,m → U be the m-th order thickening of vE ∈ U . We get induced morphisms

ιn φE jm : UE ,m →Mn . Since ÒφE : ÒU →Mn is formally smooth, this compatible system lifts

via ÒφE to give rise to a morphism γ: ÒU → ÒU such that ÒφE γ= ιn φE η, where η: ÒU →U is

the formal completion morphism of U at vE . That is, we have a commutative diagram

ÒU ÒU

U [Un/GLn ] Mn

←

→η

←

→γ

←

→

ÒφE

←

→φE

←

→ιn

where, since ιn φE and η are isomorphisms to first order at vE (i.e. at the level of tangent

spaces), the same is true for γ, so γ is an isomorphism.

The inclusion ÒU ⊂ ÒV is cut out by quadrics (the derivatives of the cubic function ga ) and

thus γ extends to an isomorphism γ: ÒV → ÒV . Let h = bga γ so that obviously (dh ) = (dbga ). The

commutativity of the diagram, along with Equation (3.25), implies that

η∗φ∗E ι∗n (s

dern ) = γ∗Òφ∗E (s

dern ) = h + (dh )2 ∈H0(S0

ÒU ).

After possible further shrinking of V around vE , we have

(3.26) φ∗E ι∗n (s

dern ) = k + (dk )2

for some function k : V →C satisfying (dk ) = (dga |V ). Therefore, taking completions, we find

η∗φ∗E ι∗n (s

dern ) = bk + (dbk )2 ∈H0(S0

ÒU ).

Since (dh ) = (dbk ) = (dbga ) it follows that bk has no quadratic terms and we must have bk −h ∈

(dbga )2. The ideal (dbga )

2 is generated by quartics so by comparing cubic terms we deduce that

bk3 = bga dγ= h3

where bk3, h3 are the cubic terms of bk , h respectively. As k is a polynomial, we get for the higher

order terms

bk −h = bk≥4−h≥4 ∈ (dbga )2

h≥deg k ∈ (dbga )2.

Since by definition h = bga γ, where bga is a cubic, we may truncate the power series defining

γ at sufficiently high degree to get a polynomial isomorphism bρ : ÒV → ÒV such that h ′ = bga bρstill satisfies (dh ′) = (dh ) = (dbga ) and h ′−h consists of summands of h of degree at least degk .

By the above, h −h ′ ∈ (dbga )2 and therefore

(3.27) η∗φ∗E ι∗n (s

dern ) = bk + (dbk )2 = h + (dh )2 = h ′+ (dh ′)2 ∈H0(S0

ÒU ).

ButV is an open subscheme of an affine space and hence bρ is the completion of an isomor-

phism ρ : V → V . In particular, h ′ is the completion of the function g ′a = ga |V ρ : V →C,

(3.28) h ′ =cg ′a : ÒV →C.

Writing I = (dga |V ) = (dg ′a ), the commutative diagram

0 SU OV /I 2ΩV /I ·ΩV

0 SU OV /I 2ΩV /I ·ΩV

←

→

←

→

⇐

⇐

←

→d

←

→ ρ∗

←

→ ρ∗

←

→

←

→

←

→d


and (3.24) show that

(3.29) φ∗E (scritn ) = g ′a + (dg ′a )

2 ∈H0(S0U ).

Using equations (3.26), (3.27), (3.28) and (3.29), the following proposition applied to the

functions k , g ′a : V → C shows that there exists a Zariski open neighbourhood vE ∈ U ′ ⊂U

and a smooth morphism

φ′E : U ′→UφE−→ [Un/GLn ]

such that

(φ′E )∗s crit

n = (φ′E )∗ι∗n s der

n ∈H0(S0U ′ ).

As [E ] varies inMn , the morphismsφ′E give a smooth surjective cover ofMn , on which the

d-critical structures s critn and ι∗n s der

n agree. Hence we must have s critn = ι∗n s der

n , as desired.

Proposition 3.11. Let V be a smooth scheme and let f1, f2 : V → C be two functions with(d f1) = (d f2) = I such that U = crit( f1) = crit( f2) ⊂ V . Fix a point u ∈ U . Let η: ÒU → U bethe formal completion of U at the point u and bI = I ⊗OV

OÒV the ideal of ÒU in ÒV .Suppose that

(3.30) f1+ bI 2 = f2+ bI 2 ∈H0(S0ÒU ).

Then there exists a Zariski open neighbourhood V ′ ⊂ V of u in V such that for U ′ =U ×V V ′

we have

f1

V ′+ I

2V ′= f2

V ′+ I

2V ′∈H0(S0

U ′ ).

Proof. Since the problem is local in V , we may assume that V is affine. Write m for the max-

imal ideal of the closed point 0 ∈ ÒV and mu for the ideal of u ∈ V . We use η: ÒV → V to

denote the formal completion map as well. Let Vloc denote the localisation of V at u . The

map η clearly factors through the localisation morphism Vloc → V by a local, faithfully flat

morphism µ: ÒV → Vloc.

Let δ = f1 − f2 : OV ,u → OV ,u/I 2. The assumption (3.30) implies that µ∗δ = 0. Therefore,

since µ is faithfully flat, we must have δ= 0.

Hence there exist functions h ∈ I 2, g ∈OV ,u \mu and a positive integer N such that

g N ( f1− f2) = h ∈ I 2.

Letting V ′ be the non-vanishing locus of g completes the proof.

3.5. Some remarks on the proof of Theorem 2.2. We finally collect some observations to

conclude the section and suggest an alternative, stronger approach to prove Theorem 2.2.

One can check that the classical truncation of the isomorphism

ιn :Spec (Q3)n/GLn

Mn

←

→∼

given in (3.13) is the isomorphism

ιn : [Un/GLn ] Mn .←

→∼

The target Mn of ιn admits the −1-shifted symplectic structureωn .

On the other hand, using the model for shifted symplectic forms for (affine) derived quo-

tient stacks described in [23, Section 1], we can construct an explicit −1-shifted symplectic

form ω = (ω0, 0, 0, . . .) onSpec (Q3)n/GLn

, combining the −1-shifted symplectic forms for


Rcrit( fn ) and B GLn , such that the induced d-critical structure on the truncation [Un/GLn ] is

s critn .

Theorem 2.2 will then follow from the stronger statement that ι∗nωn =ω as−1-shifted sym-

plectic forms.

While we certainly believe this to be true, we have chosen to pursue a local argument to

obtain the equality of d-critical structures using the derived deformation theory of Mn . This

gives us more control over the derived tangent complex and explicit Darboux charts, which we

are comfortable with, and also avoids possible technicalities on shifted symplectic structures

on derived quotient stacks.

4. THE D-CRITICAL STRUCTURE(S) ON QuotA3 (O ⊕r , n )

Fix integers n ≥ 0 and r ≥ 1. Recall from Theorem 2.3 the critical structure

ncQuotnr ⊃ crit( fr,n ) Qr,n =QuotA3 (O ⊕r , n )←

→∼

on the Quot scheme Qr,n of length n quotients O ⊕r ։ E of the trivial rank r sheaf on A3. In

this section, we compare two d-critical structures

s critr,n ∈H0

S

0crit( fr,n )

, s der

r,n ∈H0S

0Qr,n

,

by bootstrapping the arguments in our discussion of the d-critical structure of Mn in the

preceding section.

To motivate, let Qr,n be the derived Quot scheme [13]. Then we have a commutative dia-

gram

(4.1)

Qr,n Qr,n

Mn Mn

←- →jr,n

←

→qr,n

←

→ q r,n

←- →jn

where jr,n and jn are the inclusions of the underlying classical spaces and qr,n is the forgetful

morphism taking [O ⊕r ։ E ] ∈Qr,n to [E ] ∈Mn , which can be viewed as the truncation of its

derived enhancement q r,n .

4.1. The d-critical structure s critr,n coming from quiver representations. The d-critical struc-

ture

s critr,n = fr,n + (d fr,n )

2

on crit( fr,n ) ,→ ncQuotnr admits a local description similar to that of s crit

n . We fix some notation

for convenience, extending the corresponding notation used for Mn in Section 3.1.

Let Yr,n = Rep(1,n )(eL3) = EndC(C

n )3 ⊕HomC(C,Cn )r = Yn ⊕Zr,n , i.e. we denote by Zr,n the

space of r -tuples (v1, . . . , vr ) of vectors vi ∈ Cn . Let fr,n : Yr,n → A

1 be the regular (and GLn -

invariant) function induced by the potential W = A[B , C ]. It is crucial to observe that this

function does not interact with the component Zr,n . According to Theorem 2.3, there is a


stability condition θ on eL3 along with a commutative diagram

Y θ−str,n Yr,n

Yr,n θ GLn ncQuotnr A1

←- →open

←

→/GLn

←

→ fr,n

⇐

⇐

←

→fr,n

defining the noncommutative Quot scheme, on which the function fr,n descends, and there

is an isomorphism ιr,n : crit( fr,n ) e→Qr,n . For instance, one could take θ = (n ,−1) for any fixed

n > 0.

Let a = (a1, . . . , ak ) denote a k -tuple of positive integers such that∑

1≤i≤k ai = n . Denote

by Q the quiver

. . .

... ...v (1)1

v (1)r

v (k )1

v (k )r

∞

1 k

B1

C1A1

Bk

CkAk

so that

Yr,a =Rep(1,a1,...,ak )(Q ) =

k∏

i=1

Yr,ai=

k∏

i=1

Rep(1,ai )(eL3).

Consider the stability condition θ = (n ,−1, . . . ,−1) on Q , so that a Q -representation with di-

mension vector (1, a1, . . . , ak ) has slope 0. We then have the following.

Lemma 4.1. Set θi = (ai ,−1) and θ = (n ,−1, . . . ,−1). Then there are open immersions

Y θ−str,a ,→

k∏

i=1

Y θi−str,ai

,→ Yr,a .

Proof. By the very definition of Yr,a , only the first inclusion needs a proof. Set d = (1, a1, . . . , ak )

and note that d · θ = 0. Any d -dimensional Q -representation M can be written in the form

M = (M1, . . . , Mk ), where Mi ∈ Rep(1,ai )(eL3). We want to show that if such an M is θ -stable,

then each Mi is θi -stable. So let us assume by contradiction that there exists 0 6=Ni (Mi such

that (dimNi ) · (ai ,−1) ≥ 0 for some 1 ≤ i ≤ k . Now, write dimNi = (d∞, di ) where 0 ≤ d∞ ≤ 1

and 0≤ di ≤ ai , so the inequality we are assuming is d∞ai −di ≥ 0, from which it follows that

d∞ = 1 (otherwise Ni = 0), and hence di < ai (otherwise Ni =Mi ).


Consider now the proper nonzero subrepresentation N = (M1, . . . , Mi−1, Ni , Mi+1, . . . , Mk )⊂

M , whose dimension vector is (1, a1, . . . , ai−1, di , ai+1, . . . , ak ). We have

(dimN ) ·θ = (1, a1, . . . , ai−1, di , ai+1, . . . , ak ) · (n ,−1, . . . ,−1)

= n −

∑

j 6=i

a j

!−di

=∑

j

a j −

∑

j 6=i

a j

!−di

= ai −di > 0,

contradicting stability of M . The result follows.

It is clear from the definitions and Lemma 4.1 that if a representationAi , Bi , Ci , v (i )1 , . . . , v (i )r

1≤i≤k

∈ Yr,a

is θ -stable then the n-dimensional (unframed) representation

(A1 ⊕ · · ·⊕Ak , B1⊕ · · ·⊕Bk , C1⊕ · · ·⊕Ck ) ∈Repn (L3)

is spanned by the vectors

v j =

v (1)j

v (2)j...

v (k )j

∈Cn , 1≤ j ≤ r.

There is a closed embeddingΦr,a : Yr,a ,→ Yr,n , which on Yn restricts to the embeddingΦa : Ya ,→

Yn (considered in Section 3.1) and concatenates the elements of∏k

i=1 Zr,aito produce an el-

ement of Zr,n .

The reductive algebraic group GLa =∏k

i=1 GLaiacts on Yr,a by componentwise conjuga-

tion on the summand Ya and Φr,a is equivariant with respect to the inclusion GLa ⊂ GLn by

block diagonal matrices of the same kind.

On the space Yr,a we have the GLa -invariant potential

(4.2) g r,a = fr,a1⊕ · · ·⊕ fr,ak

: Yr,a →A1, (Ai , Bi , Ci , Zi )i 7→

k∑

i=1

Tr Ai [Bi , Ci ],

where (Ai , Bi , Ci ) ∈ Yaiand Zi = (v

(i )1 , . . . , v (i )r ) ∈ Zr,ai

=HomC(C,Cai )r for i = 1, . . . , k . We have

the obvious relation fr,n Φr,a = g r,a , and we observed above that the θ -stable locus embeds

in the θ -stable locus, which yields a commutative diagram

Y θ−str,a Y θ−st

r,n

Yr,a Yr,n A1

←- →

←-

→ ←-

→

← →g r,a

←- →Φr,a ←

→fr,n

where the vertical arrows are open immersions and θ = (n ,−1).

If we set

Ur,n = crit( fr,n )⊂ Yr,n , Ur,a = crit(g r,a )⊂ Yr,a ,


the restriction Φr,a : Ur,a ,→Ur,n is still equivariant with respect to the inclusion GLa ⊂ GLn ,

and so it induces a morphism of schemes

(4.3) ψr,a : Ur,a θ GLa →Ur,n θ GLn ,

which fits in a diagram

Ur,a θ GLa

∏ki=1 Ur,ai

θiGLai

∏ki=1 Qr,ai

Ur,n θ GLn Qr,n

←- →open

←

→ψr,a

←

→∼

←

→←

→∼

Let Qr,a ,→Qr,a =∏k

i=1 Qr,aibe the open subscheme of k -tuples of quotients

Ka = ([O⊕r ։ E1], . . . , [O ⊕r ։ Ek ]) ∈Qr,a

such that Supp(Ei )∩ Supp(E j ) = ; for all i 6= j . We can use the above diagram to identify the

mapψr,a with the ‘union of points’ map (denoted the same way)

ψr,a : Qr,a →Qr,n ,

which takes a point Ka ∈Qr,a as above to the joint surjection [O ⊕r ։ E1⊕· · ·⊕Ek ]∈Qr,n . This

morphism is étale by [1, Proposition A.3].

Let K = [O ⊕r ։ E ] ∈ Qr,n be in the image of the map ψr,a , i.e. assume we can write

K =ψr,a (Ka ). The local structure of the d-critical locus (crit( fr,n ), s critr,n ) around ι−1

r,n (K ) is then

described as follows.

Lemma 4.2. If φr,a denotes the composition

φr,a : U θ−str,a Ur,a θ GLa Ur,n θ GLn ,←

→

←

→ψr,a

then we have an identity of d-critical structures

φ∗r,a s critr,n = g r,a + (dg r,a )

2 ∈H0(S0U θ−st

r,a)GLa .

Proof. The commutative diagram

Ur,a θ GLa Yr,a θ GLa

Ur,n θ GLn Yr,n θ GLn A1

←- →

←

→ψr,a

←

→Φr,a

←

→

g r,a

←- →

←

→fr,n

and the fact that ψr,a is étale immediately imply the claim, using the defining properties of

the sheaf S0U θ−st

r,a.

4.2. The d-critical structure s derr,n coming from derived symplectic geometry. We start right

away with the following definition.

Definition 4.3. The derived d-critical structure on Qr,n is defined as

s derr,n = q ∗r,n s der

n ∈H0S

0Qr,n

.


Remark 4.4. We observed in Theorem 2.6 how Theorem 3.10, combined with Proposition 2.5,

proves the relation


r,n ∈H0S

0crit( fr,n )

that is the content of Theorem A.

In the rest of this subsection, we motivate and justify the above definition by explaining how

the d-critical structure s derr,n arises naturally using the derived Quot scheme in Diagram (4.1).

We proceed to describe the derived Quot scheme Qr,n .

Recall that as in the case of classical Quot schemes [22], the derived Quot scheme Qr,n is

defined as follows: LetA3 ⊂ P be any compactification ofA3, for exampleP= P3. Then Qr,n is

the open derived subscheme of the derived Quot scheme QP,r,n [13] parametrising quotients

[O ⊕rP ։ E ]where E is a 0-dimensional length n sheaf on P whose support is contained inA3.

By direct computation, we have that Qr,n is a quasi-affine dg-scheme (or more generally

derived scheme) [Spec R/GLn ] where R is a sheaf of commutative dg-algebras generated in

degrees 0,−1,−2 respectively by

OY θ−str,n

, Repn (L3)⊕3 ≃ gl⊕3

n , Repn (L3)≃ gln

and the differentials are exactly the same as for the algebra (Q3)n . Notice that we are slightly

abusing the notation Spec R here.

It is now clear that there is an inclusion (Q3)n ⊂R of dg-algebras, where (Q3)n is as in Equa-

tion (3.13). This induces precisely the forgetful map

q r,n : Qr,n −→Mn .

Since the map of dg-algebras is evidently smooth, it follows that q r,n is smooth.

Given these descriptions of Qr,n , Mn and the simple “product" structure of the map q r,n ,

we see that while the derived Quot scheme is not−1-shifted symplectic, any Darboux chart of

Mn will give rise to a smooth chart of Qr,n which is a smooth d-critical chart for the classical

truncation Qr,n . These d-critical charts induce precisely the d-critical structure s derr,n of Defini-

tion 4.3. The (closed, degenerate) 2-form q ∗r,nωn is compatible with this d-critical structure

in the appropriate sense.

5. THE CASE OF A COMPACT CALABI–YAU 3-FOLD

Let F be a locally free sheaf of rank r on a smooth, projective Calabi–Yau 3-fold X , where

we have fixed a trivialisation ζ: Λ3ΩX e→OX . Form the Quot scheme

QF,n =QuotX (F, n ).

In this section we prove Theorem B, showing that the derived d-critical structure on QF,n , de-

fined in (5.1) below, is locally modelled on the derived critical structure s derr,n of Definition 4.3.

We will need the following algebraic result.

Lemma 5.1. Let x ∈ X be a point on a smooth projective Calabi–Yau 3-fold X , and let 0 ∈A3

be the origin. There is an equivalence of cyclic dg-algebras

Ext∗(Ox ,Ox )∼= Ext∗(O0,O0).

Moreover the higher Massey products mn vanish for n ≥ 3.


Proof. Let X be any smooth 3-fold, x ∈ X a point. The cyclic dg-algebra Ext∗(Ox ,Ox ) can be

computed in either the algebraic or analytic category, the result and its A∞-structure being

the same, capturing the deformation theory of a point inside a smooth 3-fold. The first state-

ment then follows after identifying suitable analytic neighbourhoods of x ∈ X and 0 ∈ A3.

The vanishing of the Massey products mn : Ext1(Ox ,Ox )⊗n → Ext2(Ox ,Ox ) for n ≥ 3 is a conse-

quence of Lemma 3.6.

Let now x be a point on a Calabi–Yau 3-fold X . We let

(−,−)x : Ext j (Ox ,Ox )×Ext3− j (Ox ,Ox )→ Ext3(Ox ,Ox )tr−→C

be the Serre duality pairing. The cyclic structure of the Ext algebra is encoded in the relations

(m2(a1, a2), a3)x = (m2(a2, a3), a1)x

for all ai ∈ Ext1(Ox ,Ox ), where m2 agrees with the Yoneda product. The diagram

Ext j (Ox ,Ox )×Ext3− j (Ox ,Ox ) Ext3(Ox ,Ox ) C

Ext j (O0,O0)×Ext3− j (O0,O0) Ext3(O0,O0) C

←→ ∼

←

→

←→ ∼

←

→trx

←→ ∼

←

→

←

→tr0

shows that the cyclic structures agree up to a nonzero scalar.

We have a commutative diagram

QF,n MX (n )

Symn X

←→h←

→q

←→ p

where MX (n ) is the moduli stack of 0-dimensional sheaves of length n over X , the morphism

p is the map to the coarse moduli space (the n-th symmetric product Symn X = X n/Sn ) and

q = qF,n is the (smooth) forgetful morphism sending a surjection [F ։ E ] to the point [E ]. The

composition h = p q agrees with the Quot-to-Chow map [17, Section 6]. The trivialisation

ζ: Λ3ΩX e→OX induces a canonical −1-shifted symplectic structure ωX ,n on MX (n ), whose

truncation sX ,n = τ(ωX ,n ) ∈H0(S0MX (n )

) induces a d-critical structure

(5.1) sF,n = q ∗sX ,n

on the Quot scheme QF,n .

Fix a 0-cycle [E ] ∈ Symn X represented, as ever, by a polystable sheaf

(5.2) E =k⊕

i=1

Cai ⊗Oxi, xi ∈ X , xi 6= xi for i 6= j .

In this particular case, the Ext quiver QE• associated to E (see Toda’s paper [30, Section 3.3]

for a more general definition) is the quiver with vertex set V (QE• ) = 1, 2, . . . , k and edge set

E (QE• ) =∐

1≤i , j≤k

Ei , j ,

where Ei , j ⊂ Ext1(Oxi,Ox j)∨ is a C-linear basis. It follows that Ei , j = ; for i 6= j , and

(5.3) Ei ,i = ei ,1, ei ,2, ei ,3 ⊂ Ext1(Oxi,Oxi)∨


contains 3 elements. The source and target maps s and t from E (QE• ) to the vertex set V (QE• )

both send Ei ,i to the vertex i . In other words, the quiver QE• is a disjoint union of k copies of

the 3-loop quiver (see Figure 4).

1

e1,1

e1,2

e1,3

2

e2,1

e2,2

e2,3

k· · ·

ek ,1

ek ,2

ek ,3

FIGURE 4. The Ext quiver of a 0-dimensional polystable sheaf (5.2).

There is an associated convergent superpotential (see [30, Sections 2.2, 2.6] for more de-

tails)

WE• ∈CQE• ⊂CJQE•K,

whose general definition reads as follows. First, for each Oxi∈ Coh(X ), consider the Massey

products on the dg-algebra Ext∗(Oxi,Oxi), defined by the maps

mn : Ext1(Oxi,Oxi)⊗n → Ext2(Oxi

,Oxi).

As in the proof of Lemma 5.1, denote by

(−,−)xi: Ext2(Oxi

,Oxi)×Ext1(Oxi

,Oxi)→ Ext3(Oxi

,Oxi)

tr−→C

the Serre duality pairing. Then, by the Calabi–Yau condition, for any given elements a1, . . . , an ∈

Ext1(Oxi,Oxi), one has the cyclicity relation

(mn−1(a1, . . . , an−1), an )xi= (mn−1(a2, . . . , an ), a1)xi

.

Let E ∨i ,i = e ∨i ,1, e ∨i ,2, e ∨i ,3 ⊂ Ext1(Oxi,Oxi) be the dual basis of (5.3). Then, Toda defines in [30,

Section 5.5] the superpotential

WE• =∑

n≥3

∑

ψ

∑

ei∈Eψ(i ),ψ(i+1)

aψ,e• · e1 · · ·en ,

whereψ runs over the set of maps 1, 2, . . . , n +1→ 1, . . . , k such thatψ(1) =ψ(n +1), and

the coefficients aψ,e• ∈C are defined by

(5.4) aψ,e• =1

n(mn−1(e

∨1 , . . . , e ∨n−1), e ∨n ).

We now determine the (trace of the) superpotential WE• explicitly.

Lemma 5.2. Given a polystable sheaf E as in (5.2), one has, up to a scalar,

Tr WE• =

k∑

i=1

Tr Ai [Bi , Ci ],

where we have set Ai = ei ,1, Bi = ei ,2 and Ci = ei ,3. In particular, Tr WE• defines a regular(everywhere convergent) function on Repa (QE• ), where a = (a1, . . . , ak ) is determined by (5.2).

Proof. By Lemma 5.1, mn = 0 for n ≥ 3, so by (5.4) only n = 3 contributes to the sum. This

already proves the last statement, about convergence. Since the only nonvanishing Ext groups


Ext1(Oxi,Ox j) are those where i = j , everyψ in the sum satisfiesψ(i ) =ψ(i +1), hence the sum

overψ is actually a sum over integers from 1 up to k . Now, we have relations

ji = (m2(A∨i , B∨i ), C ∨i )xi

= (m2(B∨i , C ∨i ), A∨i )xi

= (m2(C∨i , A∨i ), B∨i )xi

li = (m2(A∨i , C ∨i ), B∨i )xi

= (m2(C∨i , B∨i ), A∨i )xi

= (m2(B∨i , A∨i ), C ∨i )xi

.

Thus

WE• =

k∑

i=1

ji

3(Ai Bi Ci +Bi Ci Ai +Ci Ai Bi ) +

li

3(Ai Ci Bi +Bi Ai Ci +Ci Bi Ai ).

But since m2 agrees with the Yoneda pairing, we have

ji + li = (m2(A∨i , B∨i ), C ∨i )xi

+ (m2(B∨i , A∨i ), C ∨i )xi

= (m2(A∨i , B∨i ) +m2(B

∨i , A∨i ), C ∨i )xi

= 0.

Since ji and li do not depend on i , we can set j = ji and l = li , so that l =− j , thus

WE• =j

3

k∑

i=1

Ai [Bi , Ci ]+Bi [Ci , Ai ]+Ci [Ai , Bi ].

It follows that Tr WE• = j ·∑

1≤i≤k Tr Ai [Bi , Ci ], as required.

Let Ma (QE• ) denote the moduli stack of a -dimensional representations of the Ext quiver

QE• . In the diagram

A1 Repa (QE• ) Ma (QE• )∏

1≤i≤k Mai(L3)

Ma (QE• )∏

1≤i≤k Symai A3

←

→TrWE• ←

→

←

→π

←→

⇐

⇐

⇐

⇐

we thus have a canonical identification (Repa (QE• ), TrWE• ) = (Ya , ga ), where ga was defined in

Equation (3.2) and it equals Tr WE• by Lemma 5.2.

By [29, Theorem 5.3], we can find open analytic neighbourhoods

0∈ V ⊂Ma (QE• ), [E ] ∈T ⊂ Symn X

and an analytic isomorphism i fitting in a diagram

Za =crit(ga |V )

GLa

p−1(T ) MX (n ),

←

→i∼ ←- →

open

with the crucial property that

(5.5) sX ,n

Za= ga |V + (dga |V )

2 ∈ H0(S0Za),

where we are abusing notation and writing ga |V for the restriction of ga toπ−1(V )⊂Repa (QE• ).

Now form the open subscheme

QF,a =Za ×MX (n )QF,n ,→QF,n ,


and consider the cartesian diagram

(5.6)

QF,a QF,n

Z ′a Za MX (n )

Qr,n Mn

←→ q

←- →

←

→ q

←

→étale

←

→

←→ ψa

←- →

←

→qr,n

defining the scheme Z ′a . Note that, possibly after shrinking V , thanks to Lemma 3.2 and

Lemma 3.3 we may assumeψa : Za →Mn is étale and satisfies

(5.7) ψ∗a s dern = ga |V + (dga |V )

2.

Next, we show that the morphisms

(5.8) QF,a →Za , Z′a →Za

look the same étale locally. We work locally analytically around a point [E ] ∈Za ⊂MX (n ). We

let [E ′]∈Mn be the image of [E ] under the étale mapψa . Let

B ⊂Cr n

be the analytic open subset corresponding to surjective maps in

HomX (F, E ) =HomA3 (O ⊕r , E ′) =Cr n .

We identify B with the fibre of q (resp. qr,n ) over the point [E ] (resp. [E ′]). Since the mor-

phisms (5.8) are smooth, they are both analytically locally trivial (on the source), so any point

in q−1([E ])⊂QF,a admits an analytic open neighbourhood of the form B×Wa , where Wa ⊂Za

is a suitable analytic open neighbourhood of [E ]. Repeating the same reasoning with the map

Z ′a → Za and shrinking Wa further if necesssary, we see that in Diagram (5.6) we can make

the replacement

QF,a

Z ′a Za

←→ q

←

→

B ×Wa

B ×Wa Wa

←→ pr2

←

→pr2

where B ×Wa has an étale map (the sameψa as above) down to Qr,n . Now we compute

sF,n

B×Wa

= pr∗2sX ,n

Wa

= pr∗2ga |V + (dga |V )

2

by (5.5)

= pr∗2ψ∗a s der

n by (5.7)

= s dern

B×Wa

.

This completes the proof of Theorem B.


6. THE SPECIAL CASE OF THE HILBERT SCHEME

This section is devoted to the proof of Theorem C.

Let H =Hilbn A3 be the Hilbert scheme parametrising 0-dimensional subschemes ofA3 of

length n . Denote by

0→ IZ →OA3×H→OZ → 0

the universal short exact sequence living over A3 ×H, where Z ⊂ A3 ×H denotes the univer-

sal subscheme. Let π: A3 ×H → H be the projection; we use the notation RHomπ(−,−) =

Rπ∗RHom (−,−) throughout.

Consider the derived critical locus Q=Rcrit( f )where f = f1,n : ncQuotn1 →A

1 is the poten-

tial in Theorem 2.3. More precisely, Q = Spec B where B is the sheaf of dg-algebras which is

generated by B 0 =OncQuotn1

in degree 0 and B−1 = TncQuotn1

in degree −1 with differential given

by the dual of the section d f ∈H0(ΩncQuotn1). Recall that ncQuotn

1 = Y θ−st1,n /GLn .

Then we have the following commutative diagram

(6.1)

H crit( f ) Q

Mn Mn

←

→ q

←

→ε∼

←

→

η

←- →j

←

→ q

←- →jn

where we have set ε = ι−11,n , and q = q1,n is the forgetful map, whereas j and jn are the inclu-

sions of the classical spaces into their derived enhancements.

The morphism q is obtained as follows: Recall that by (3.13) Mn is the quotient stack

[Spec (Q3)n/GLn ]. For brevity, let us write A = (Q3)n and A0 = (Q3)0n = OYn

, A−1 = (Q3)−1n = TYn

be the degree 0 and degree−1 summands of the dg-algebra (Q3)n respectively. There are natu-

ral morphisms A0→ B 0 and A−1→ B−1 which are GLn -invariant and, together with the trivial

map A−2→ 0, induce a morphism Spec B → Spec A→ [Spec A/GLn ], which is the morphism

q .

Note that, by definition,

Ecrit = j ∗LQ Lcrit( f )

←

→ϕcrit

is the critical obstruction theory on crit( f ), denoted Ef in the introduction, see (0.7). The

maps η and q induce a commutative diagram

η∗q ∗LMn

ε∗Ecrit η∗LQ

LH

←→

ψ ←

→

⇐

⇐

←

→ε∗ϕcrit


which after applying the truncation functor τ[−1,0] becomes

(6.2)

η∗τ[−1,0]q∗LMn

ε∗Ecrit

LH

←

→

ψ ←

→

ϕ

←

→ε∗ϕcrit

whereEcrit is unchanged since it already has cohomological amplitude in degrees [−1, 0]. The

morphism ψ is an isomorphism because it is the pullback along η of the second of the iso-

morphisms (6.3) established in the following lemma.

Lemma 6.1. The derivative of q induces isomorphisms

(6.3)

TQ τ[0,1]q∗TMn

τ[−1,0]q∗LMn

LQ.

←

→∼

←

→∼

Moreover, one has an isomorphism

(6.4) γ: τ[0,1]RHomπ(OZ [−1],OZ ) τ[0,1]η∗q ∗TMn

.←

→∼

Proof. The derived tangent complex of Q is given by

TQ = [TncQuotn1

ΩncQuotn1]

←

→Hess( f )

where Hess( f ) denotes the Hessian of f .

Since ncQuotn1 = Y θ−st

1,n /GLn and the action of GLn on Y θ−st1,n is free, we may write this as the

quasi-isomorphic GLn -equivariant three-term complex on Y θ−st1,n in degrees−1 to 1 (where we

are slightly abusing notation by omitting certain pullbacks)

TQ = [gln TY θ−st1,n

ΩncQuotn1].←

→

←

→Hess( f )

By the definition of q , we have (again slightly abusing notation)

q ∗TMn= [gln TYn

ΩYngln ]

←

→

←

→

←

→

and the derivative dq is the morphism of complexes

gln TY θ−st1,n

ΩncQuotn1

gln TYnΩYn

gln

←

→

⇐⇐

←

→Hess( f )

←→ ←→

←

→

←

→

←

→

We may identify TY θ−st1,n

with (gl⊕3n ⊕C

n )⊗OB and TYnwith gl⊕3

n ⊗OB so that the middle vertical

arrow is the natural projection map.

It is clear that H0(dq ) is surjective.

To show that it is injective, notice that the leftmost arrow gln → TY θ−st1,n

maps X ∈ gln to

([X , A], [X , B ], [X , C ], X v ) ∈ gl⊕3n ⊕C

n at the point (A, B , C , v ) ∈ Y θ−st1,n . Since, by stability, v is a

cyclic vector with respect to the action of the matrices A, B , C , for any w ∈ Cn there exists a

polynomial f (A, B , C ) such that w = f (A, B , C )v . Letting X = f (A, B , C ), the image of X is then


(0, 0, 0, w ) ∈ gl⊕3n ⊕C

n . Therefore, the composition gln → TY θ−st1,n= gl⊕3

n ⊕Cn → Cn is fibrewise

surjective and hence is a surjective morphism of locally free sheaves and the injectivity of

H0(dq ) follows readily.

A similar argument involving duals shows that H1(dq ) is an isomorphism as well.

Thus the truncation τ[0,1]dq induces the first isomorphism in (6.3). The second isomor-

phism is obtained analogously.

For (6.4), since the universal complex of Mn over the image of q restricts to OZ on Mn ,

we have that RHomπ(OZ [−1],OZ ) is isomorphic to q ∗ j ∗nTMn= η∗q ∗TMn

, so applying the

truncation τ[0,1] gives the desired isomorphism.

We note that by the above explicit description ofTMnand the morphism q we have an iso-

morphism between the complexes η∗τ[0,1]q∗TMn

and τ[0,1]η∗q ∗TMn

, which we use to iden-

tify the two complexes from now on.

Our final goal is to produce an isomorphism

ρ : η∗τ[−1,0]q∗LMn

Eder =RHomπ(IZ ,IZ )0[2]

←

→∼

such that

(6.5)


Eder

LH

←

→ρ

←

→ϕ←

→ ϕder

commutes, whereϕder is obtained from the Atiyah class of IZ via a classical construction (see

[19], [24] or [25] for full details).

First of all, we have a diagram

0 Rπ∗OA3×H[1] Rπ∗OA3×H[1]

RHomπ(IZ ,OZ ) RHomπ(IZ ,IZ )[1] RHomπ(IZ ,OA3×H)[1]

RHomπ(IZ ,OZ ) RHomπ(IZ ,IZ )0[1] RHomπ(OZ ,OA3×H)[2]

←

→

←

→

⇐

⇐

←

→

←

→

←

→

←

→id

←

→

←

→

←

→

←

→

←

→

in the derived category of H, where rows and columns are exact triangles; more precisely:

(1) the middle column is obtained by applying Rπ∗ to the shifted dual of the exact triangle

RHom (IZ ,IZ )0 → RHom (IZ ,IZ ) → OA3×H , exploiting the fact that all three objects

are self-dual;

(2) the right column is obtained by applying RHomπ(−,OA3×H) to the exact triangleOZ [−2]→

IZ [−1]→OA3×H[−1],

(3) the middle row is obtained by applying RHomπ(IZ ,−) to the exact triangle OZ →

IZ [1]→OA3×H[1].

The last row induces an isomorphism

α: τ[0,1]RHomπ(IZ ,OZ ) RHomπ(IZ ,IZ )0[1].

←

→∼


On the other hand, the exact triangle

OA3×H[−1] OZ [−1] IZ

←

→

←

→

induces, via RHomπ(−,OZ ), an exact triangle

RHomπ(IZ ,OZ ) RHomπ(OZ ,OZ )[1] RHomπ(OA3×H ,OZ )[1]

←

→

←

→

such that the truncation of the first arrow

β : τ[0,1]RHomπ(IZ ,OZ ) τ[0,1]RHomπ(OZ ,OZ )[1]

←

→∼

is an isomorphism. Summing up, as proved in [11, Proposition 2.2], we have an isomorphism

β α−1 : RHomπ(IZ ,IZ )0[1] τ[0,1]RHomπ(OZ [−1],OZ )

←

→∼

along with the identifications


=τ[0,1]η

∗q ∗TMn

∨

=τ[0,1]RHomπ(OZ [−1],OZ )

∨via γ∨ from (6.4)

= (RHomπ(IZ ,IZ )0[1])∨ via (β α−1)∨

= Eder by Serre duality.

We have thus obtained

ρ = (β α−1)∨ γ∨ : η∗τ[−1,0]q∗LMn

Eder

←

→∼

Thus all that remains to check is the commutativity of Diagram (6.5); that is, we need to check

that “up to ρ” the vertical map

ϕ : η∗τ[−1,0]q∗LMn

LH

←

→

in Diagram (6.2) agrees with the symmetric obstruction theoryϕder : Eder→LH on the Hilbert

scheme, viewed as a moduli space of ideal sheaves.

Recall that the Atiyah class of a perfect complex P on a scheme Y is an element

AtP ∈ Ext1(P, P ⊗LY [1]) =Hom(P [−1], P ⊗LY ).

Working over Y =A3×H, by projecting along πwe will view our Atiyah classes as elements in

the group

AtP ∈Hom(P [−1], P ⊗π∗LH).

By the functoriality of Atiyah classes, we have a commutative diagram

OA3×H[−1] OZ [−1] IZ

OA3×H ⊗π∗LH OZ ⊗π

∗LH IZ [1]⊗π∗LH

←

→

←

→ 0=AtO

←

→

←→ AtOZ

←

→

u ←

→ AtIZ

←

→

←

→

where we observed that the Atiyah class of the trivial line bundle vanishes, to give the diagonal

arrow

u : IZ →OZ ⊗π∗LH .


The commutativity of the two triangles forming the right square translates into a commutative

diagram

RHom (IZ ,IZ [1])∨

RHom (IZ ,OZ )∨ π∗LH

RHom (OZ [−1],OZ )∨

←

→OZ→IZ [1]

←

→

AtIZ

←

→u

←

→

AtOZ

←

→

OZ [−1]→IZ

After dualising and applying τ[0,1] Rπ∗, we obtain a commutative diagram

RHomπ(IZ ,IZ )0[1]

τ[0,1]RHomπ(IZ ,OZ ) Rπ∗π∗TH TH

τ[0,1]RHomπ(OZ [−1],OZ )

τ[0,1]η∗q ∗TMn

←

→

α

←

→β

←

→

←

→

←

→

ϕ∨der

←

→

ϕ∨

←

→

π∗At∨OZ

←

→γ

where γ is the isomorphism (6.4), and the lower right part of the diagram, stating that the

composition γ π∗At∨OZ agrees with the map ϕ∨ dual to the map ϕ appearing in Diagram

(6.2), is an immediate consequence of [26, Appendix A] or [18, Proposition 2.4.7].

Dualising back, we conclude the proof of Theorem C. Therefore we have proved Conjecture

9.9 in [14].

REFERENCES

1. Sjoerd Beentjes and Andrea T. Ricolfi, Virtual counts on Quot schemes and the higher rank local DT/PT corre-

spondence, To appear in Math. Res. Lett. ArXiv:1811.09859, 2018.

2. Kai Behrend, Introduction to Donaldson–Thomas theory, II, MSRI Talk, 2018.

3. Kai Behrend and Barbara Fantechi, The intrinsic normal cone, Invent. Math. 128 (1997), no. 1, 45–88.

4. Kai Behrend and Barbara Fantechi, Symmetric obstruction theories and Hilbert schemes of points on threefolds,

Algebra Number Theory 2 (2008), 313–345.

5. Oren Ben-Bassat, Christopher Brav, Vittoria Bussi, and Dominic Joyce, A ‘Darboux theorem’ for shifted sym-

plectic structures on derived Artin stacks, with applications, Geom. Topol. 19 (2015), no. 3, 1287–1359.

6. Yuri Berest, George Khachatryan, and Ajay Ramadoss, A Simple Construction of Derived Representation

Schemes, ArXiv:1010.4901, 2010.

7. Christopher Brav, Vittoria Bussi, Delphine Dupont, Dominic Joyce, and Balázs Szendroi, Symmetries and sta-

bilization for sheaves of vanishing cycles, J. Singul. 11 (2015), 85–151, With an appendix by Jörg Schürmann.

8. Christopher Brav, Vittoria Bussi, and Dominic Joyce, A Darboux theorem for derived schemes with shifted sym-

plectic structure, J. Amer. Math. Soc. 32 (2019), no. 2, 399–443.

9. Christopher Brav and Tobias Dyckerhoff, Relative Calabi–Yau structures II: Shifted Lagrangians in the moduli

of objects, To appear in Selecta Math. ArXiv:1812.11913, 2018.

10. Vittoria Bussi, Dominic Joyce, and Sven Meinhardt, On motivic vanishing cycles of critical loci, J. Algebraic

Geom. 28 (2019), no. 3, 405–438.

https://arxiv.org/abs/1811.09859

https://www.msri.org/workshops/815/schedules/23405




11. Yalong Cao and Martijn Kool, Counting zero-dimensional subschemes in higher dimensions, J. Geom. Phys.

136 (2019), 119–137.

12. Alberto Cazzaniga, Dimbinaina Ralaivaosaona, and Andrea T. Ricolfi, Higher rank motivic Donaldson–

Thomas invariants of A3 via wall-crossing, and asymptotics, ArXiv:2004.07020, 2020.

13. Ionut Ciocan-Fontanine and Mikhail Kapranov, Derived Quot schemes, Ann. Sci. École Norm. Sup. (4) 34

(2001), no. 3, 403–440.

14. Nadir Fasola, Sergej Monavari, and Andrea T. Ricolfi, Higher rank K-theoretic Donaldson–Thomas theory of

points, Forum Math. Sigma 9 (2021), no. E15, 1–51.

15. Victor Ginzburg, Calabi–Yau algebras, ArXiv:0612139, 2006.

16. Alexander Grothendieck, Éléments de Géométrie Algébrique (rédigés avec la collaboration de Jean Dieudonné):

III. Étude cohomologique des faisceaux cohérents, Seconde partie, Inst. Hautes Études Sci. Publ. Math. (1963),

no. 32, 5–91.

17. Alexander Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les

schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Soc. Math. France, Paris, 1995, pp. Exp. No. 221, 249–276.

18. Benjamin Hennion, Tangent Lie algebra of derived Artin stacks, J. Reine Angew. Math. 741 (2018), 1–45.

19. Daniel Huybrechts and Richard P. Thomas, Deformation-obstruction theory for complexes via Atiyah and

Kodaira-Spencer classes, Math. Ann. 346 (2010), no. 3, 545–569.

20. Dominic Joyce, A classical model for derived critical loci, J. Differ. Geom. 101 (2015), no. 2, 289–367.

21. Young-Hoon Kiem and Michail Savvas, K-Theoretic Generalized Donaldson–Thomas Invariants, Int. Math. Res.

Not. IMRN (2020) Online.

22. Nitin Nitsure, Construction of Hilbert and Quot schemes, Fundamental Algebraic Geometry, Math. Surveys

Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 105–137.

23. Tony Pantev, Bertrand Toën, Michel Vaquié, and Gabriele Vezzosi, Shifted symplectic structures, Publ. Math.

Inst. Hautes Études Sci. 117 (2013), 271–328.

24. Andrea T. Ricolfi, Virtual classes and virtual motives of Quot schemes on threefolds, Adv. Math. 369 (2020),

107182.

25. Andrea T. Ricolfi, The equivariant Atiyah class, C. R. Math. Acad. Sci. Paris 359 (2021), no. 3, 257–282.

26. Timo Schürg, Bertrand Toën, and Gabriele Vezzosi, Derived algebraic geometry, determinants of perfect com-

plexes, and applications to obstruction theories for maps and complexes, J. Reine Angew. Math. 702 (2015),

1–40.

27. The Stacks Project Authors, Stacks Project, stacks-project, 2021.

28. Balázs Szendroi, Non-commutative Donaldson–Thomas invariants and the conifold, Geom. Topol. 12 (2008),

no. 2, 1171–1202.

29. Yukinobu Toda, Gopakumar–Vafa invariants and wall-crossing, ArXiv:1710.01843, 2017.

30. Yukinobu Toda, Moduli stacks of semistable sheaves and representations of Ext-quivers, Geom. Topol. 22 (2018),

no. 5, 3083–3144.

31. Bertrand Toën, Higher and derived stacks: a global overview, Algebraic geometry—Seattle 2005. Part 1, Proc.

Sympos. Pure Math., vol. 80, Amer. Math. Soc., Providence, RI, 2009, pp. 435–487.

32. Bertrand Toën and Michel Vaquié, Moduli of objects in dg-categories, Ann. Sci. École Norm. Sup. (4) 40 (2007),

no. 3, 387–444.

33. Bertrand Toën and Gabriele Vezzosi, From HAG to DAG: derived moduli stacks, Axiomatic, enriched and mo-

tivic homotopy theory, NATO Sci. Ser. II Math. Phys. Chem., vol. 131, Kluwer Acad. Publ., Dordrecht, 2004,

pp. 173–216.

Andrea T. Ricolfi, [email protected]

SCUOLA INTERNAZIONALE SUPERIORE DI STUDI AVANZATI (SISSA), VIA BONOMEA 265, 34136

TRIESTE, ITALY

Michail Savvas, [email protected]

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF CALIFORNIA, SAN DIEGO, LA JOLLA, CA 92093,

USA


https://arxiv.org/abs/math/0612139

https://academic.oup.com/imrn/advance-article-abstract/doi/10.1093/imrn/rnaa097/5840461

http://stacks.math.columbia.edu


thed-criticalstructure onthequotschemeofpointsofacalabi

Documents