arXiv:0810.5645v6 [math.AG] 7 Jul 2010 arXiv:0810.5645v6 [math.AG] 7 Jul 2010...

download arXiv:0810.5645v6 [math.AG] 7 Jul 2010 arXiv:0810.5645v6 [math.AG] 7 Jul 2010 AtheoryofgeneralizedDonaldson–Thomas

of 212

  • date post

    02-Aug-2020
  • Category

    Documents

  • view

    4
  • download

    0

Embed Size (px)

Transcript of arXiv:0810.5645v6 [math.AG] 7 Jul 2010 arXiv:0810.5645v6 [math.AG] 7 Jul 2010...

  • ar X

    iv :0

    81 0.

    56 45

    v6 [

    m at

    h. A

    G ]

    7 J

    ul 2

    01 0

    A theory of generalized Donaldson–Thomas

    invariants

    Dominic Joyce∗ and Yinan Song†

    Abstract

    Donaldson–Thomas invariants DTα(τ ) are integers which ‘count’ τ - stable coherent sheaves with Chern character α on a Calabi–Yau 3-fold X, where τ denotes Gieseker stability for some ample line bundle on X. They are unchanged under deformations of X. The conventional defini- tion works only for classes α containing no strictly τ -semistable sheaves. Behrend showed that DTα(τ ) can be written as a weighted Euler char- acteristic χ

    (

    M α st(τ ), νMαst(τ)

    )

    of the stable moduli scheme Mαst(τ ) by a constructible function νMαst(τ) we call the ‘Behrend function’.

    This book studies generalized Donaldson–Thomas invariants D̄Tα(τ ). They are rational numbers which ‘count’ both τ -stable and τ -semistable coherent sheaves with Chern character α on X; strictly τ -semistable sheaves must be counted with complicated rational weights. The D̄Tα(τ ) are defined for all classes α, and are equal to DTα(τ ) when it is defined. They are unchanged under deformations of X, and transform by a wall- crossing formula under change of stability condition τ .

    To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be writ- ten locally as Crit(f) for f : U → C holomorphic and U smooth, and use this to deduce identities on the Behrend function νM. We compute our invariants D̄Tα(τ ) in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories mod-CQ/I of representations of a quiver Q with relations I coming from a superpotential W on Q, and connect our ideas with Szendrői’s noncom- mutative Donaldson–Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontsevich and Soibelman’s independent paper [63].

    Contents

    1 Introduction 3 1.1 Brief sketch of background from [49–55] . . . . . . . . . . . . . . 5 1.2 Behrend functions of schemes and stacks, from §4 . . . . . . . . . 7

    ∗Address: The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, U.K., E-mail: joyce@maths.ox.ac.uk

    †E-mail: yinansong@gmail.com

    1

    http://arxiv.org/abs/0810.5645v6

  • 1.3 Summary of the main results in §5 . . . . . . . . . . . . . . . . . 8 1.4 Examples and applications in §6 . . . . . . . . . . . . . . . . . . 10 1.5 Extension to quivers with superpotentials in §7 . . . . . . . . . . 12 1.6 Relation to the work of Kontsevich and Soibelman [63] . . . . . 13

    2 Constructible functions and stack functions 15 2.1 Artin stacks and (locally) constructible functions . . . . . . . . . 16 2.2 Stack functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 Operators Πµ and projections Πvin . . . . . . . . . . . . . . . . . . 20 2.4 Stack function spaces S̄F, S̄F(F, χ,Q) . . . . . . . . . . . . . . . . 22

    3 Background material from [51–54] 24 3.1 Ringel–Hall algebras of an abelian category . . . . . . . . . . . . 24 3.2 (Weak) stability conditions on A . . . . . . . . . . . . . . . . . . 26 3.3 Changing stability conditions and algebra identities . . . . . . . . 29 3.4 Calabi–Yau 3-folds and Lie algebra morphisms . . . . . . . . . . 31 3.5 Invariants Jα(τ) and transformation laws . . . . . . . . . . . . . 34

    4 Behrend functions and Donaldson–Thomas theory 35 4.1 The definition of Behrend functions . . . . . . . . . . . . . . . . . 36 4.2 Milnor fibres and vanishing cycles . . . . . . . . . . . . . . . . . . 38 4.3 Donaldson–Thomas invariants of Calabi–Yau 3-folds . . . . . . . 43 4.4 Behrend functions and almost closed 1-forms . . . . . . . . . . . 44 4.5 Characterizing Knum(coh(X)) for Calabi–Yau 3-folds . . . . . . . 46

    5 Statements of main results 50 5.1 Local description of the moduli of coherent sheaves . . . . . . . . 53 5.2 Identities on Behrend functions of moduli stacks . . . . . . . . . 60 5.3 A Lie algebra morphism, and generalized D–T invariants . . . . . 61 5.4 Invariants counting stable pairs, and deformation-invariance . . . 65

    6 Examples, applications, and generalizations 69 6.1 Computing PIα,n(τ ′), D̄Tα(τ) and Jα(τ) in examples . . . . . . 69 6.2 Integrality properties of the D̄Tα(τ) . . . . . . . . . . . . . . . . 76 6.3 Counting dimension zero sheaves . . . . . . . . . . . . . . . . . . 79 6.4 Counting dimension one sheaves . . . . . . . . . . . . . . . . . . 80 6.5 Why it all has to be so complicated: an example . . . . . . . . . 85 6.6 µ-stability and invariants D̄Tα(µ) . . . . . . . . . . . . . . . . . 90 6.7 Extension to noncompact Calabi–Yau 3-folds . . . . . . . . . . . 90 6.8 Configuration operations and extended D–T invariants . . . . . . 95

    7 Donaldson–Thomas theory for quivers with superpotentials 97 7.1 Introduction to quivers . . . . . . . . . . . . . . . . . . . . . . . . 98 7.2 Quivers with superpotentials, and 3-Calabi–Yau categories . . . . 101 7.3 Behrend functions, Lie algebra morphisms, and D–T invariants . 106 7.4 Pair invariants for quivers . . . . . . . . . . . . . . . . . . . . . . 108

    2

  • 7.5 Computing D̄TdQ,I(µ), D̂T d

    Q,I(µ) in examples . . . . . . . . . . . 115

    7.6 Integrality of D̂TdQ(µ) for generic (µ,R,6) . . . . . . . . . . . . . 122

    8 The proof of Theorem 5.3 129

    9 The proofs of Theorems 5.4 and 5.5 131 9.1 Holomorphic structures on a complex vector bundle . . . . . . . 132 9.2 Moduli spaces of analytic vector bundles on X . . . . . . . . . . 136 9.3 Constructing a good local atlas S for M near [E] . . . . . . . . . 137 9.4 Moduli spaces of algebraic vector bundles on X . . . . . . . . . . 139 9.5 Identifying versal families of vector bundles . . . . . . . . . . . . 140 9.6 Writing the moduli space as Crit(f) . . . . . . . . . . . . . . . . 142 9.7 The proof of Theorem 5.4 . . . . . . . . . . . . . . . . . . . . . . 144 9.8 The proof of Theorem 5.5 . . . . . . . . . . . . . . . . . . . . . . 145

    10 The proof of Theorem 5.11 147 10.1 Proof of equation (5.2) . . . . . . . . . . . . . . . . . . . . . . . . 147 10.2 Proof of equation (5.3) . . . . . . . . . . . . . . . . . . . . . . . . 150

    11 The proof of Theorem 5.14 153

    12 The proofs of Theorems 5.22, 5.23 and 5.25 162 12.1 The moduli scheme of stable pairs Mα,nstp (τ

    ′) . . . . . . . . . . . . 162 12.2 Pairs as objects of the derived category . . . . . . . . . . . . . . 165 12.3 Cotangent complexes and obstruction theories . . . . . . . . . . . 166 12.4 Deformation theory for pairs . . . . . . . . . . . . . . . . . . . . 168 12.5 A non-perfect obstruction theory for Mα,nstp (τ

    ′)/U . . . . . . . . . 171 12.6 A perfect obstruction theory when rankα 6= 1 . . . . . . . . . . . 176 12.7 An alternative construction for all rankα . . . . . . . . . . . . . 179 12.8 Deformation-invariance of the PIα,n(τ ′) . . . . . . . . . . . . . . 182

    13 The proof of Theorem 5.27 183 13.1 Auxiliary abelian categories Ap,Bp . . . . . . . . . . . . . . . . . 184 13.2 Three weak stability conditions on Bp . . . . . . . . . . . . . . . 188 13.3 Stack function identities in SFal(MBp) . . . . . . . . . . . . . . . 190

    13.4 A Lie algebra morphism Ψ̃Bp : SFindal (MBp) → L̃(Bp) . . . . . . . 194 13.5 Proof of Theorem 5.27 . . . . . . . . . . . . . . . . . . . . . . . . 195

    Glossary of Notation 204

    Index 209

    1 Introduction

    Let X be a Calabi–Yau 3-fold over the complex numbers C, and OX(1) a very ample line bundle on X . Our definition of Calabi–Yau 3-fold requires X to be

    3

  • projective, with H1(OX) = 0. Write coh(X) for the abelian category of coherent sheaves on X , and Knum(coh(X)) for the numerical Grothendieck group of coh(X). We use τ to denote Gieseker stability of coherent sheaves with respect to OX(1). If E is a coherent sheaf on X then the class [E] ∈ Knum(coh(X)) is in effect the Chern character ch(E) of E in Heven(X ;Q).

    For α ∈ Knum(coh(X)) we form the coarse moduli schemes Mαss(τ),M α st(τ)

    of τ -(semi)stable sheaves E with [E]=α. Then Mαss(τ) is a projective C-scheme whose points correspond to S-equivalence classes of τ -semistable sheaves, and Mαst(τ) is an open subscheme of M

    α ss(τ) whose points correspond to isomor-

    phism classes of τ -stable sheaves. For Chern characters α with Mαss(τ) = M

    α st(τ), following Donaldson and

    Thomas [20, §3], Thomas [100] constructed a symmetric obstruction theory on Mαst(τ) and defined the Donaldson–Thomas invariant to be the virtual class

    DTα(τ) = ∫ [Mαst(τ)]

    vir 1 ∈ Z, (1.1)

    an integer which ‘counts’ τ -semistable sheaves in class α. Thomas’ main re- sult [100, §3] is that DTα(τ) is unchanged under deformations of the underly- ing Calabi–Yau 3-fold X . Later, Behrend [3] showed that Donaldson–Thomas invariants can be written as a weighted Euler characteristic

    DTα(τ) = χ ( Mαst(τ), νMαst(τ)

    ) , (1.2)

    where νMαst(τ) is the Behrend function, a constructible function on M α st(τ) de-

    pending only on Mαst(τ) as a C-scheme. (Here, and throughout, Euler charac- teristics are taken with respect to cohomology with compact support.)

    Conventional Donaldson–Thomas invariants DTα(τ) are only defined for classes α with Mαss(τ) = M

    α st(τ), that is, when there are no strictly