ATIYAH CLASS AND SHEAF COUNTING ON LOCAL CALABI YAU FOURFOLDS D

FOURFOLDS

DUILIU-EMANUELDIACONESCU AND ARTANSHESHMANI AND SHING-TUNG

YAU

Abstract

We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 di- mensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the sur- face. We prove that in certain cases, when the rank 2 bundle is chosen appro- priately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as tor- sion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class en- ables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface us- ing results in [15,16]. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.

CONTENTS

1. Introduction 2

1.1. General results 3

1.2. Rank two results and conjectures for elliptic K3 surfaces 5

Aknowledgement 6

2. Universal relative Atiyah classes on product schemes 7

2.1. Truncated Atiyah class of a scheme 7

2.2. Atiyah classes on product schemes 8

2.3. Proof of Theorem 2.5 18

3. Obstruction theory for sheaves on local Calabi-Yau varieties 18

3.1. Comparison of relative Atiyah classes 19

3.2. Proof of Proposition 3.2. 24

3.3. Comparison of obstruction theories 25

4. Donaldson-Thomas invariants for sheaves on local fourfolds 27

1

4.1. Stable two dimensional sheaves 28

4.2. Reduced obstruction theory 30

4.3. Reduced Donaldson-Thomas invariants 31

5. Rank two invariants and universality 32

5.1. Rank two fixed loci 32

5.2. Universality results 35

6. Rank two calculations for local elliptic K3 surfaces 38 6.1. Chamber structure for local elliptic surfaces 38 6.2. Type II fixed loci for elliptic K3 surfaces 43 6.3. Partition functions and modularity conjectures 45

Appendix A. Background results 47

A.1. Adjunction 47

A.2. Products and diagonals 48

References 51

1. Introduction

This paper is motivated by two recent developments in Donaldson-Thomas the- ory, namely the mathematical theory of Vafa-Witten invariants of complex sur- faces constructed in [31, 32], [16, 15] and [19], and the construction [3, 4, 5] of virtual cycle invariants for sheaves on Calabi-Yau fourfolds. In the approach of [15,31] the Vafa-Witten invariants of a smooth projective surfaceSare constructed as reduced virtual cycle invariants of two dimensional sheaves on the total space of the canonical bundle KS. In fact, the construction of loc. cit. is more general, including equivariant residual invariants of two dimensional sheaves on the total spaceYof an arbitrary line bundleLonS. The starting point for the present study is the observation that the total space X of the rank two bundle L⊕K_S ⊗L⁻¹ is a Calabi-Yau fourfold, which is at the same time isomorphic to the total space of the canonical line bundle KY. This leads to the natural question whether the in- variants constructed in [15, 31] for Y are related to fourfold Donaldson-Thomas invariants. In fact such a correspondence has already been proven in [4] for the case whereY is a compact Fano threefold, using a local Kuranishi map construc- tion for fourfold Donaldson-Thomas invariants. The main goal of this paper is prove an analogous correspondence for more general situations allowingYto be a quasi-projective threefold with a suitable torus action. As discussed in detail below, the approach employed here is different, using the truncated Atiyah class formalism of [27]. A second goal is to test the modularity conjecture for two di- mensional sheaf counting invariants in the local fourfold framework. This will be

carried out for a particular class of examples, whereYis the total space of certain line bundlesLover an elliptic K3 surfaceS.

The main motivation for this study resides in modular properties of Donaldson- Thomas invariants for sheaves with two dimensional support. This has been a central theme in enumerative geometry starting with the early work of [17,38,39, 40,42,18] on torsion free sheaves on surfaces. It has been also studied intensively for two dimensional sheaves on Calabi-Yau threefolds [13,12,36,14,34,35]. Fur- ther results on this topic include [19,22,23,31]. As modularity is a natural conse- quence of S-duality [37,7,6,11], similar physical arguments predict that it should be also manifest in the Donaldson-Thomas theory of two dimensional sheaves on Calabi-Yau fourfolds. At the moment this question is completely open. This paper essentially consists of two parts, as explained below.

1.1. General results.LetYbe a smooth quasi-projective variety of dimensiond− 1, d ≥ 2, and let X be the total space of the canonical bundle K_Y. Let M be a quasi-projective fine moduli scheme of compactly supported stable sheavesF on X. Suppose that all sheaves, F, parameterized by M are scheme theoretically supported on Y. The formalism of truncated Atiyah classes developed in [27], yields two natural (unreduced) “deformation-obstruction theories”,aX : E^•X →_L^•_M and a_Y : E^•Y → _L^•_M for the moduli space M. These are associated respectively to deformations of sheaves, realized, respectively, as modules over the structure rings of X and Y. As in [2], in the present paper an obstruction theory will be an object E^• of amplitude [−1, 0] in the derived category of the moduli space, together with a morphism in derived category φ : E^• → _L^•_M such that h⁰(φ) is an isomorphism andh⁻¹(_φ)is an epimorphism of sheaves. HereL^•_Mdenotes the [−1, 0]truncation of the full cotangent complex of Mas in [27]. The first general result proven in this paper is

Theorem 1.1. (Proposition 3.2 and Proposition 3.6). There is a natural splitting in D^b(M):E^•X ∼=E^•Y⊕_E^•∨_Y[1−d]. Moreover, the obstruction theory aXfactors through E^•Y and there is a direct sum decomposition

aX = (a_Y, 0).

This shows that, under the above assumptions, the fourfold obstruction the- ory ofM admits a reduction to the threefold obstruction theory. Although this is a natural statement, the proof of it, using the truncated Atiyah classes, is sur- prisingly long. In particular, it uses an explicit calculation, Theorem 2.5, for the relative Atiyah class of a product.

Using this result, the proof of Theorem 1.1 is then given subsequently in Sec- tions 3.1 and 3.3, the main intermediate results being Propositions 3.2 and 3.6

respectively. In particular, Propositions 3.6 is a more explicit restatement of Theo- rem 1.1.

Applying Theorem 1.1, a reduced equivariant Donaldson-Thomas theory is con- structed in Section 4 for two dimensional sheaves on certain local fourfolds. In sec- tion 4, S is a smooth projective surface with nef canonical bundle, K_S, equipped with a polarization h = c1(O_S(1)). It is also assumed that H¹(O_S) = 0 and the integral cohomology ofSis torsion free. Given an effective divisorDonS, a four- fold X is constructed as the total space of a rank two bundle KS(D)⊕ O_S(−_D). The threefold Y is the total space of the line bundle L = K_S(D) _on S, which is canonically embedded inXas a divisor.

As in [15] the topological invariants of a compactly supported two dimensional sheafF on X are completely determined by the total Chern class c(g∗F) of the direct imageg∗F, whereg: X→Sis the canonical projection. In turn, the latter is naturally given by a triple

(1) γ(_F) = (r(_F),β(_F),n(_F))∈ _Z⊕H₂(S,Z)⊕_Z

The entry,r(F) = rk(g∗F), will be called the rank ofF in the following. More- over, sinceDis effective nonzero divisor, one easily shows (Corollary 4.3) that any suchh-stable sheafF is scheme theoretically supported onY ⊂ X. This yields a natural identification of the moduli spacesM_h(X,γ) ∼= M_h(Y,γ). Moreover, as proven in [15, Proposition 2.4] and [31, Theorem 6.5] the latter has areducedperfect obstruction theorya_Y^red : E^•red_Y →_L^•_M. The detail of this construction is reviewed in Section 4.2 for completeness. Then Theorem 1.1 implies:

Corollary 1.2. The fourfold obstruction theory aX : E^•X → _L^•_M admits a reduc- tion to the reduced perfect obstruction theory a^red_Y : E^•red_Y → _L^•_M constructed in [15, Proposition 2.4]and[31, Theorem 6.5].

Using this result, the reduced equivariant Donaldson-Thomas invariants for two dimensional sheaves onX are defined by residual virtual integration in Sec- tion 4.3. The next two sections, 5 and 6, are focused on explicit computations and modularity conjectures for rank two invariants. As shown in [15, Section 3] and [31, Section 7], there are two types of rank two torus fixed loci. The “type I” fixed locus is isomorphic to the moduli space M_h(S,γ) of torsion free sheaves on S with topological invariants γ. The “type II” fixed loci consist of sheaves F with scheme theoretic support on the divisor 2S ⊂Y, whose connected components are naturally isomorphic to twisted nested Hilbert schemes, as reviewed in detail in Section 5.1. By analogy with similar previous results [15,19], universality results for residual virtual integrals of type I and II are proven in Propositions 5.1 and 5.3. These are valid for any pair(S,D)as above. Explicit results and a modularity conjecture are then obtained in Section 6 for elliptic K3 surfaces withDa positive

multiple of the fiber class,D =m f.

1.2. Rank two results and conjectures for elliptic K3 surfaces.LetSbe a smooth generic elliptic K3 surface in Weierstrass form. By the genericity assumption, all elliptic fibers are reduced and irreducible, and have at most nodal singularities.

Furthermore, the Mordell-Weill group is trivial and the Picard group ofSis freely generated by the fiber classF and the canonical section classσ. The K¨ahler cone ofSthen consists of real divisor classesh =tσ+u f with t,u∈ _{R, 0}<2t <u.

Let D = m f, m ≥ 1, be a positive multiple of the elliptic fiber so that the local fourfoldXis the total space ofO_S(m f)⊕ O_S(−m f). Moreover, let the topological invariant (1) be of the formγ = (2, f,n) withn ∈ Z. The Bogomolov inequality, proven in Lemma 6.6, shows that in this case the moduli spaceM_h(Y,γ)is empty forn < 0. Moreover, the global boundedness result proven in Lemma 6.7 shows that there are no marginal stability walls associated to sheavesF with invariants γfor

(2) 0 < ^t

u < ¹ 1+8n.

Therefore for any fixedn≥0 the equivariant residual Donaldson-Thomas invari- ants DT_h(X,γ) reach a stable limit, D_∞(X,γ) ∈ _Q(s) as the size of the elliptic fibers decreases, keeping the size of the section fixed. Then let

Z_∞(X, 2, f;q) =

∑

n≥0

DT_∞(X, 2, f,n)qⁿ⁻² ∈q⁻²Q(s)[[q]]

be the generating function of limit invariants. Clearly the above partition func- tions splits into type I and type II contributions denoted by Z_∞(X, 2,f;q)_I and Z_∞(X, 2, f;q)_{I I} respectively. Then the following result is proven in Section 6.3.

Proposition 1.3. The series Z_∞(X, 2, f;q)_I belongs to Q(s)[[q]] ⊂ _Q(s)[[q⁻¹,q]]

and the following identity holds in the quadratic extensionQ(s)[[q^1/2]]:

(3) Z∞(X, 2,f;q)_I = ^s

−1

2 (_∆⁻¹(q^1/2) +_∆⁻¹(−q^1/2)), where∆(q)is the discriminant modular form.

Moreover, Proposition 6.15 shows that all type II nonzero contributions vanish for m even. At the same time for m odd, all nonzero type II contributions are associated to nested Hilbert schemes without divisorial twists. Then the follow- ing conjecture is inferred from the results of [31, Section 8] using the universality result Proposition 5.3.

Conjecture 1.4. Suppose m ≥₁is odd. Then

(4) Z_∞(X, 2, f;q)_{I I} = ^s

−1

4 ∆⁻¹(q²).

Plan of the paper. The key theorem of the current article is the reduction The- orem 1.1. We approach the problem by essentially proving a statement that the universal (relative) Atiyah class of the (universal) sheaf F on the moduli space, M_h(X,γ) of stable torsion sheavesF onX, with scheme theoretic support onY, has a reduction to the universal Atiyah class of the universal sheafGonM_h(Y,γ) (which is isomorphic toM_h(X,γ)) parameterizing stable sheavesG onY(where for allF ∈ M_h(X,γ)we haveF ∼=i∗G,i: Y ,→X). This will be done in Proposi- tion 3.2. Our proof of Proposition 3.2 relies heavily on a certain splitting property of the universal relative Atiyah classes on product schemes, as stated in Theorem 2.5. Hence, we start in Section 2 a detailed discussion of Atiyah classes on product schemes which can be read independently from the rest of the paper. Then, in Sec- tion 3 we use Theorem 2.5 to prove Proposition 3.2 and later on, as an application of Proposition 3.2, we show in Proposition 3.6 that the reduction of Atiyah classes as above, leads to a reduction in the level of deformation obstruction theories, as stated in Theorem 1.1. The latter proves that in our current geometric setup the Donaldson-Thomas invariants of fourfold X can be completely recovered from those of threefold Y which, after being reduced, enjoy modularity property, dis- cussions of which we have included in Sections 4, 5 and 6.

Aknowledgement

D.-E. D would like to thank Ron Donagi and Tony Pantev for their interest in this work and very helpful discussions. D.-E.D. was partially supported by NSF grants DMS-1501612 and DMS-1802410. A.S. would like to thank Amin Gholam- pour, Alexey Bondal, Alexander Efimov for helpful discussions and commenting on first versions of this article. A.S. was partially supported by NSF DMS-1607871, NSF DMS-1306313 and Laboratory of Mirror Symmetry NRU HSE, RF Govern- ment grant, ag. No 14.641.31.0001. A.S. would like to further sincerely thank the Center for Mathematical Sciences and Applications at Harvard University, the center for Quantum Geometry of Moduli Spaces at Aarhus University, and the Laboratory of Mirror Symmetry in Higher School of Economics, Russian feder- ation, for the great help and support. S.-T. Y. was partially supported by NSF DMS-0804454, NSF PHY-1306313, and Simons 38558.

2. Universal relative Atiyah classes on product schemes

As explained in the plan of the paper, this section is a self-contained account of relative truncated Atiyah classes, the main result being Theorem 2.5 below.

2.1. Truncated Atiyah class of a scheme.Let W be an arbitrary scheme of finite type overC and W ⊂ Aa closed embedding in a smooth scheme A. As in [27, Definition. 2.1], the truncated cotangent complexLW is the two term complex of amplitude[−1, 0]

JW/J_W² →_Ω¹_A|_W,

where JW ⊂ O_A is the defining ideal sheaf of W ⊂ A. Let us denote by ∆W : W → W×W the diagonal morphism, byO_∆W the extension by zero of the struc- ture sheaf of the diagonal, and byI_∆W ⊂ O_W×W the ideal sheaf of the diagonal.

Then the following results are proven in [27, Lemma 2.2, Definition 2.3].

Lemma 2.1. (i) There is an exact sequence ofO_W×W-modules (5) 0→_∆W∗(_JW/JW²)−→I^τW _∆A|_W×W−→I^ξW _∆W →0

where the morphismξWis the natural projection.

(ii) LetιW : I_∆W → O_W×W denote the canonical inclusion. Then there is a commuta- tive diagram ofO_W×W-modules

(6) ∆W∗(_JW/JW²) ^{τW //}

1

I_∆A|_W×W ^{ιW◦ξW //}

r_W

O_W×W

∆W∗(_JW/JW²) ^{∆W∗(dW/A) //}(I_∆A/I_∆²

A)|_W×W.

where the right vertical map rW is the natural projection and the bottom row is naturally isomorphic toLW[1].

By construction, the top row of diagram (6) is quasi-isomorphic toO_∆W. Hence, the Diagram (6) determines a morphismαW : O_∆W →_∆W∗LW[1]. The latter is the truncated universal Atiyah class ofW. According to [27, Definition2.1], the trun- cated cotangent complex and truncated Atiyah class are independent of choice of A. As shown in [27, Definition 2.3], an alternative presentation of the truncated Atiyah class is obtained from the proof of Lemma 2.2 in loc. cit., as follows.

Lemma 2.2. (i) There is a canonical isomorphism (7) I_∆A|_W×A ∼=I_∆W⊂W×A

where the right hand side denotes the defining ideal sheaf for the closed embedding

∆W ⊂W×A.

(ii) The isomorphism(7)determines a further canonical isomorphism ofO_W×A-modules (8) (O_WJW)⊗_W×AO_∆W−→^∼ _∆W∗(_JW/JW²)

and a natural injection

(9) O_WJW ,→ I_∆A|_W×A

which fit in a commutative diagram of morphisms ofO_W×A-modules:

(10) O_WJW  //

I_∆A|_W×A

(O_WJW)⊗_W×AO_∆W

o

ν_W //I_∆A|_W×W

r_W

∆W∗(_JW/J²_W) ^{∆W∗(dW/A) //}

τ_W

33

(I_∆A/I_∆²

A)|_W×W

Moreover, the two upper vertical arrows in the above diagram are the natural pro- jections.

Remark 2.3. By a slight abuse of notation, in the above lemma∆Windicates the canon- ical closed embedding W ,→W×A as opposed to the diagonal embedding W ,→W×W.

This notation will be consistently used throughout this section, the distinction being clear from the context.

Next note the following immediate corollary of Lemma 2.2.

Corollary 2.4. The universal Atiyah class is equivalently defined by the commutative diagram

(11) (O_WJW)⊗_W×AO_∆W ^{νW //}

1

I_∆A|_W×W ^{ιW◦ξW //}

r_W

O_W×W

(O_WJW)⊗_W×AO_∆W ^{rW◦νW //} (I_∆A/I_∆²

A)|_W×W.

where the left column is restricted to W×W. As in Lemma2.1, the right vertical map is the natural projection.

2.2. Atiyah classes on product schemes.In most applications to moduli problems W will be a product schemeZ×T, where Zis a smooth quasi-projective variety and T is a test scheme, or the moduli space. In the latter case, as shown in [27, Thm. 4.1], the construction of the obstruction theory of the moduli space, uses only the relative componentαW/Z of the Atiyah class (as will be recalled below).

The goal of the section is to prove Theorem 2.5 below, which relatesαW/Z to the Atiyah classαT ofT.

To fix the notation, for any productW₁×W₂of schemes overCletπW_i, 1≤i ≤ 2, denote the canonical projections. Moreover, for any test schemeTletZ_T = Z× T and, given a morphism of schemes f : Z₁ → Z₂, let f_T : (Z₁)_T → (Z₂)_T denote the morphism f ×1T. The subscripts will be suppressed ifT = Spec(_C). Given two complexes of sheaves C₁^•,C₂^• on W₁,W2, the tensor product π_W^∗

1C^•₁ ⊗π^∗_W

2C^•₂ will be denoted by C^•₁ C^•₂. Similarly, the direct sum π_W^∗

1C₁^•⊕π^∗_W

2C^•₂ will be denoted byC^•₁C₂^•. Some background results needed below have been recorded in the Appendix A.

LetW = ZT where Zis a smooth complex quasi-projective variety andT is an arbitrary quasi-projective scheme overC. Then there is a canonical splitting

(12) LZT ∼=ΩZLT.

In the present context this splitting follows from equation (112) proven in Corol- lary A.5, which shows that for any quasi-projective schemesW1,W2there is a nat- ural isomorphism

¯

ρW : I_∆W

1/I_∆²

W1 O_∆W

2 ⊕ O_∆W

1 I_∆W

2/I_∆²

W2

−→I∼ _∆W/I_∆²

W.

Since T is quasi-projective, there is a closed embedding T ⊂ U into a smooth quasi-projective schemeU. Hence one can pickA =Z_U in the construction of the Atiyah class, sinceZ is smooth. Then, settingW₁ = Zand W2 = U in the above equation yields the splitting (12).

The relative Atiyah class is the componentα_ZT/Z : O_∆

ZT → _∆ZT∗(π_T^∗LT[1]) of αZ_T with respect to this splitting. Using Corollary A.3, this is canonically identified with a morphism

α_ZT/Z : O_∆ZO_∆T → O_∆ZLT[1].

The notation can be preserved with no risk of confusion. Note also that the Atiyah class ofTis a morphismαT : O_∆T → _∆T∗LT[₁]. Then the main result of this section is to relate the latter two classes as follows:

Theorem 2.5. Let Z,T be quasi-projective schemes overCwith Z smooth. Then there is a relation

(13) α_ZT/Z =1_O∆

Z αT.

The proof will use diagram (11) in Corollary 2.4 and will proceed in several steps. Since T is quasi-projective, there is a closed embedding T ⊂ U with U smooth quasi-projective. Moreover, as already noted below equation (12) one can

set A = ZU in the construction of the Atiyah class since Z is smooth. Then dia- gram (11) reads

(14) (O_ZT JZT)⊗_ZT×ZU O_∆

ZT

ν_ZT

//

1

I_∆

ZU|_ZT×ZT ^{ιZT◦ξZT //}

r_ZT

O_ZT×ZT

(O_ZT JZ_T)⊗_ZT×ZU O_∆

ZT

r_ZT◦ν_ZT

//(I_∆

ZU/I_∆²

ZU)|_ZT×ZT,

where the left hand column is restricted to Z_T×Z_T. The strategy to prove Theo- rem 2.5 is to construct a suitable quasi-isomorphic model for the rows in the above diagram, making relation (13) manifest. In order to facilitate the exposition, dia- grams such as (14) will be regarded in an obvious way as quiver representations in abelian categories of coherent sheaves. A map between two diagrams will be a morphism of quiver representations i.e. a collection of morphisms between the sheaves associated to the respective nodes which intertwine between the maps belonging to the two representations. Then, as a first step, note the following consequence of Corollary A.3.

Lemma 2.6. Diagram(14)is canonically isomorphic to

(15) O_∆Z∆T∗(_JT/JT²) ^{νZT //}

1

I_∆

ZU|_ZT×ZT ^{ιZT◦ξZT //}

r_ZT

O_ZT×ZT

O_∆Z∆T∗(JT/J_T²) ^rZT

◦ν_ZT

// (I_∆

ZU/I_∆²

ZU)|_ZT×ZT

Moreover, the morphismνZT in diagram(15)is determined by the commutative diagram

(16) O_Z×Z(O_TJT) ^{ //}

I_∆

ZU|_ZT×ZU

O_∆Z∆T∗(_JT/JT²) ^{νZT //} I_∆

ZU|_ZT×ZT.

where the top horizontal arrow is the natural injection induced by the isomorphismI_∆

ZU|_ZT×ZU ∼= I_∆

ZT⊂Z_T×Z_U and the vertical arrows are natural projections.

Proof. Using equation (108) in Corollary A.3 it is straightforward to check that there is a commutative diagram of morphisms ofO_{ZT×ZU-modules}

(17) O_ZT JZT

∼ //

O_Z×Z(O_TJT)

(O_ZT JZT)⊗_ZT×ZUO_∆

ZT

∼ // O_∆Z∆T∗(_JT/JT²)

where the horizontal arrows are canonical isomorphisms and the vertical arrows are natural projections. This follows from the basic properties of tensor products using the canonical isomorphisms

JZ_T ∼=O_ZJT, (O_TJT)⊗_T×U O_∆T ∼=∆T∗(_JT/JT²)

ofO_ZU and O_T×U-modules respectively. The details will be omitted. Moreover, as shown in Lemma 2.2, the morphismνZT in diagram (14) is determined by the commutative diagram

(18) O_ZT JZT

 //

I_∆

ZU|_ZT×ZU

(O_ZT JZT)⊗_ZT×ZU O_∆

ZT

ν_ZT

//I_∆

ZU|_ZT×ZT

where the top horizontal arrow is the natural injection while the vertical arrows are the natural projections. Diagram (16) follows immediately from (18) using

(17). q.e.d.

For the next step, recall the splitting (12) is determined by the canonical isomor- phism

(19) ρ_U : I_∆Z/I_∆²

ZO_∆U ⊕ O_∆ZI_∆U/I_∆²

U

−→I∼ _∆

ZU/I_∆²

ZU

obtained in Corollary A.5. As also shown in loc. cit., there is a commutative diagram

(20) I_∆Z/I_∆²

ZO_∆U ⊕ O_∆ZI_∆U/I_∆²

U

ρ_U

//

(0,1)

I_∆

ZU/I_∆²

ZU

γ_ZU

O_∆ZI_∆U/I_∆²

U

1 //O_∆ZI_∆U/I_∆²

U

where γ_ZU : I_∆

ZU/I_∆²

ZU → O_∆Z I_∆U/I_∆²

U is determined by the natural projec- tionγZ_U : I_∆

ZU → O_∆ZI_∆U.

Now let

p_ZT =γ_ZU|_ZT×ZT :(I_∆

ZU/I_∆²

ZU)|_ZT×ZT → O_∆Z(I_∆U/I_∆²

U)|_T×T. and letq_ZT : I_∆

ZU|_ZT×ZT → O_∆Z(I_∆U/I_∆²

U)|_T×T be the composite map I_∆

ZU|_ZT×ZT ^{rZT //}

q_ZT **

(I_∆

ZU/I_∆²

ZU)|_ZT×ZT

p_ZT

O_∆Z (I_∆U/I_∆²

U)|_T×T.

HererZ_T is the natural projection used in diagram (15). Then note that the canon- ical projectionLZ_T → O_ZLT obtained from the splitting (12) is determined by the commutative diagram

O_∆Z∆T∗(_JT/JT²)

1

r_ZT◦ν_ZT

// (I_∆

ZU/I_∆²

ZU)|_ZT×Z_T p_ZT

O_∆Z∆T∗(_JT/JT²) ^{qZT◦νZT //}O_∆Z(I_∆U/I_∆²

U)|_T×T. As a result, Lemma 2.6 immediately implies:

Corollary 2.7. The relative Atiyah classαZ_T/Zis determined by the commutative dia- gram

(21) O_∆Z∆T∗(_JT/JT²) ^{νZT //}

1

I_∆

ZU|_ZT×ZT ^{ιZT◦ξZT //}

q_ZT

O_ZT×ZT

O_∆Z∆T∗(JT/J_T²) ^qZT

◦ν_ZT

// O_∆Z (I_∆U/I²_∆

U)|_T×T.

Note also that the Atiyah class ofTis obtained by settingZ =Spec(_C)in Corol- lary 2.4. This yields the following commutative diagram

(22) ∆T∗(_JT/JT²) ^{νT //}

1

I_∆U|_T×T ιT◦_ξT

//

r_T

O_T×T

∆T∗(_JT/JT²) ^{rT◦νT //}I_∆U/I_∆²

U|_T×T.

where the bottom row is isomorphic toLT[1]. The map νT is determined by the commutative diagram

(23) O_TJT  //

I_∆U|_T×U

∆T∗(_JT/J²_T) ^{νT //} I_∆U|_T×T.

where the top horizontal map is the natural inclusion induced by the isomorphism I_∆U|_T×U ∼=I_∆T⊂T×U, and the vertical maps are natural projections.

As the next step in the proof of Theorem 2.5, it will be shown below in Lemma 2.9 that the domain of the relative Atiyah class is quasi-isomorphic to the three term complex

(24) O_∆Z∆T∗(_JT/JT²) ^1νT// O_∆ZI_∆U|_T×T ^{1(ιT◦ξT) //}O_∆ZO_T×T.

This is an important step in the proof since it provides a direct connection with the domain of theα_T in diagram (23).

The starting point is the exact sequence ofO_ZU×ZU-modules (25) 0→ I_∆ZO_U×U

j_ZU

−−→ I_∆

ZU

γ_ZU

−−→ O_∆ZI_∆U →0,

which follows from Corollary A.4. The key idea is to gain some control over the restrictionI_∆

ZU|_ZT×ZT by proving:

Lemma 2.8. The exact sequence(25)remains exact upon restriction to Z_T×Z_T. Proof. First note that Corollary A.4 yields a second exact sequence ofO_ZU×ZU- modules

(26) 0→ I_∆Z I_∆U −−→ I^δZU _∆ZI_∆U −−→ I^ρZU _∆

ZU →0.

As shown below, this sequence remains exact upon restriction to ZT ×ZT. The restriction of (26) toZT×_ZT is the exact sequence

(27) I_∆Z (I_∆U|_T×T) → I_∆Z(I_∆U|_T×T) → I_∆ZU|_ZT×Z_T →0.

The second component of the first map from the left is

(28) ιZ¹: I_∆Z I_∆U|_T×T → O_Z×ZI_∆U|_T×T

where ιZ : I_∆Z → O_Z×Z is the canonical injection. This is injective, as shown in Lemma A.2 hence (27) is also left exact.

Now, the restriction of (25) toZT×_ZT is the exact sequence (29) I_∆ZO_T×T

−→Iφ _∆

ZU|_T×T−→O_∆ZI_∆U|_T×T →0.

where φ = j_ZU|_ZT×ZT. Furthermore, by construction there is a commutative dia- gram

I_∆ZO_T×T ^{1 //}

1 0

!

I_∆ZO_T×T

φ

0 ^//I_∆Z(I_∆U|_T×T) ^//I_∆Z(I_∆U|_T×T) ^// I_∆ZU|_T×T //0.

where the bottom row is the restriction of (26) toZT×ZT. Since the latter is exact, using equation (28), the snake lemma yields a long exact sequence

0 →Ker(φ) → I_∆Z(I_∆U|_T×T) −−−→ O^ιZ1 _Z×Z(I_∆U|_T×T)→ · · · .

As noted above,ιZ¹is injective, hence (29) is also exact on the left. q.e.d.

Now recall that the domain of the Atiyah class in diagram (21) is the complex O_∆Z∆T∗(JT/J_T²)−−→ I^νZT _∆

ZU|_ZT×ZT −−−−→ O^ιZT◦ξZT _Z×ZO_T×T. Let

(30) 0→ I_∆ZO_T×T −−→ I^eZT _∆

ZU|_ZT×ZT −−→ O^ηZT _∆ZI_∆U|_T×T →₀ denote the restriction of (25) toZ_T×Z_T and let

(31) 0 → I_∆Z−→O^ιZ _Z×Z−→O^fZ _∆Z →_0.

be the canonical exact sequence of the diagonal inZ×Z.

As claimed above equation (24), the next result proves that the domain of the relative Atiyah class is indeed quasi-isomorphic to the complex (24).

Lemma 2.9. There is a commutative diagram (32)

0

0

I_∆Z O_T×T

e_ZT

1 //I_∆ZO_T×T

i_Z¹

O_∆Z∆T∗JT/J_T²

1

ν_ZT

//I_∆

ZU|_ZT×Z_T η_ZT

ι_ZT◦ξ_ZT

//O_Z×ZO_T×T

fZ¹

O_∆Z∆T∗(_JT/JT²) ^1νT// O_∆Z I_∆U|_T×T

1(ιT◦ξT)

//

O_∆ZO_T×T

0 0

with exact columns.

Proof. The middle column is the exact sequence (30). The exactness of the right column follows from the exactness of (31) using Lemma A.2. The lower right square is commutative since all the arrows in it are natural restrictions.

In order to prove that the lower left square is commutative, note that the maps νZ_T,νT are determined by the commutative diagrams (17) and (23) respectively.

These are displayed again below for convenience:

(33) O_Z×Z(O_TJT) ^{ kZT //}

a_ZT

I_∆

ZU|_ZT×ZU

b_ZT

O_∆Z∆T∗(_JT/JT²) ^{νZT //} I_∆

ZU|_ZT×ZT

O_TJT  ^kT //

a_T

I_∆U|_T×U

b_T

∆T∗(_JT/J²_T) ^{νT //}I_∆U|_T×T

In each case the vertical arrows are canonical projections while the upper hori- zontal arrow is a natural injection. Obviously, the second‘ diagram in (33) yields

a commutative diagram ofO_{ZT×ZU-modules} (34) O_∆Z(O_TJT) ^{ 1kT //}

1^aT

O_∆Z(I_∆U|_T×U)

1^bT

O_∆Z ∆T∗(_JT/JT²) ^{1νT //}O_∆Z(I_∆U|_T×T)

Then the from Diagram (34) and the left diagram in (33) one obtains a diagram of the form

(35)

O_Z×Z(O_TJT) ^{ kZT //}

a_ZT

f_Z¹

''

I_∆

ZU|_ZT×ZU

b_ZT

^γZU|_ZT×ZU

zz

O_∆Z ∆T∗(_JT/JT²)

1

++

ν_ZT

// I_∆

ZU|_ZT×ZT

η_ZT

tt

O_∆Z(O_TJT) ^{ 1kT //}

1^aT

O_∆Z(I_∆U|_T×U)

1^bT

O_∆Z ∆T∗(_JT/JT²) ^{1νT //}O_∆Z(I_∆U|_T×T) whereγU : I_∆

ZU → O_∆Z I_∆U is the projection in (25). Since kZ_T,kT are natural injections, the following relations hold

γZ_U|_ZT×ZU ◦kZ_T = (1kT)◦(fZ¹)

Moreover, sinceηZT =_γU|_ZT×ZT and b_ZT,b_T are natural projections, one also has (1b_T)◦_γU|_ZU×ZT =_ηZT ◦b_ZT.

SinceaT,aZ_T are surjections, and two straight squares in the diagram (35) are com- mutative, this implies

ηZ_T ◦νZ_T = (1νT).

Finally, in order to prove that the upper right square is commutative, first note that Lemma A.4 yields an exact sequence

0→ I_∆ZO_T×T δ_ZT

−−→ I_∆

ZT

γ_ZT

−−→ O_∆Z I_∆T →0.