Chapter 1 Introduction ......................................... 1
1.1 Brief sketch of background from [49, 50, 51, 52, 53, 54,
55] ........................................................ 2
1.2 Behrend functions of schemes and stacks, from chapter 4 .... 4
1.3 Summary of the main results in chapter 5 ................... 5
1.4 Examples and applications in chapter 6 ..................... 7
1.5 Extension to quivers with superpotentials in chapter 7 ..... 8
1.6 Relation to the work of Kontsevich and Soibelman [63] ...... 9
Chapter 2 Constructible functions and stack functions ......... 13
2.1 Artin stacks and (locally) constructible functions ........ 13
2.2 Stack functions ........................................... 15
2.3 Operators Πμ and projections Πvin .......................... 17
2.4 Stack function spaces  .......................... 18
Chapter 3 Background material from [51, 52, 53, 54] ........... 21
3.1 Ringel-Hall algebras of an abelian category ............... 21
3.2 (Weak) stability conditions on  ......................... 23
3.3 Changing stability conditions and algebra identities ...... 25
3.4 Calabi-Yau 3-folds and Lie algebra morphisms .............. 27
3.5 Invariants Jα(τ) and transformation laws .................. 29
Chapter 4 Behrend functions and Donaldson-Thomas theory ....... 31
4.1 The definition of Behrend functions ....................... 31
4.2 Milnor fibres and vanishing cycles ........................ 33
4.3 Donaldson-Thomas invariants of Calabi-Yau 3-folds ......... 38
4.4 Behrend functions and almost closed 1-forms ............... 39
4.5 Characterizing Anum(coh(Χ)) for Calabi-Yau 3-folds ......... 40
Chapter 5 Statements of main results .......................... 45
5.1 Local description of the moduli of coherent sheaves ....... 47
5.2 Identities on Behrend functions of moduli stacks .......... 53
5.3 A Lie algebra morphism  , and
generalized Donaldson-Thomas invariants  α(τ) ........... 54
5.4 Invariants РIα,n(τ') counting stable pairs, and
deformation-invariance of the α(τ) ..................... 58
Chapter 6 Examples, applications, and generalizations ......... 63
6.1 Computing РIα,n(τ'), α(τ) and Jα(τ) in examples ......... 63
6.2 Integrality properties of the α(τ) ..................... 69
6.3 Counting dimension zero sheaves ........................... 71
6.4 Counting dimension one sheaves ............................ 73
6.5 Why it all has to be so complicated: an example ........... 77
6.6 μ-stability and invariants α(μ) ........................ 81
6.7 Extension to noncompact Calabi-Yau 3-folds ................ 82
6.8 Configuration operations and extended Donaldson-Thomas
invariants ................................................ 86
Chapter 7 Donaldson-Thomas theory for quivers with
superpotentials ..................................... 89
7.1 Introduction to quivers ................................... 89
7.2 Quivers with superpotentials, and 3-Calabi-Yau
categories ................................................ 92
7.3 Behrend function identities, Lie algebra morphisms, and
Donaldson-Thomas type invariants .......................... 96
7.4 Pair invariants for quivers ............................... 99
7.5 Computing dQ,I(μ),  dQ,I(μ) in examples ................ 104
7.6 Integrality of dQ(μ) for generic (μ , ≤) ............... 111
Chapter 8 The proof of Theorem 5.3 ........................... 119
Chapter 9 The proofs of Theorems 5.4 and 5.5 ................. 123
9.1 Holomorphic structures on a complex vector bundle ........ 124
9.2 Moduli spaces of analytic vector bundles on X ............ 127
9.3 Constructing a good local atlas S for  near [E] ........ 128
9.4 Moduli spaces of algebraic vector bundles on X ........... 130
9.5 Identifying versal families of holomorphic structures
and algebraic vector bundles ............................. 131
9.6 Writing the moduli space as Crit(ƒ) ...................... 133
9.7 The proof of Theorem 5.4 ................................. 135
9.8 The proof of Theorem 5.5 ................................. 135
Chapter 10 The proof of Theorem 5.11 ......................... 139
10.1 Proof of equation (5.2) ................................. 139
10.2 Proof of equation (5.3) ................................. 142
Chapter 11 The proof of Theorem 5.14 ......................... 147
Chapter 12 The proofs of Theorems 5.22, 5.23 and 5.25 ........ 155
12.1 The moduli scheme of stable pairs  α,nstp(τ') ........... 155
12.2 Pairs as objects of the derived category ................ 157
12.3 Cotangent complexes and obstruction theories ............ 158
12.4 Deformation theory for pairs ............................ 160
12.5 A non-perfect obstruction theory for α,nstp(τ')/U ...... 163
12.6 A perfect obstruction theory when rank α ≠ 1 ............ 167
12.7 An alternative construction for all rank α .............. 170
12.8 Deformation-invariance of the PIα,n(τ') ................. 173
Chapter 13 The proof of Theorem 5.27 ......................... 175
13.1 Auxiliary abelian categories p,  p ..................... 175
13.2 Three weak stability conditions on p ................... 179
13.3 Stack function identities in SFα(p) .................. 181
13.4 A Lie algebra morphism  ......... 184
13.5 Proof of Theorem 5.27 ................................... 186
Bibliography .................................................. 187
Glossary of Notation .......................................... 193
Index ......................................................... 197
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