# A Theory of Generalized Donaldson-Thomas Invariants

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*Dominic Joyce; Yinan Song*

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

To prove all this, the authors study the local structure of the moduli stack
\(\mathfrak M\) of coherent sheaves on \(X\). They show that an
atlas for \(\mathfrak M\) may be written locally as
\(\mathrm{Crit}(f)\) for \(f:U\to{\mathbb C}\) holomorphic and
\(U\) smooth, and use this to deduce identities on the Behrend function
\(\nu_\mathfrak M\). They compute the invariants
\(\bar{DT}{}^\alpha(\tau)\) in examples, and make a conjecture about
their integrality properties. They also extend the theory to abelian categories
\(\mathrm{mod}\)-\(\mathbb{C}Q\backslash I\) of representations
of a quiver \(Q\) with relations \(I\) coming from a
superpotential \(W\) on \(Q\).

#### Table of Contents

# Table of Contents

## A Theory of Generalized Donaldson-Thomas Invariants

- Chapter 1. Introduction 18 free
- 1.1. Brief sketch of background from [49-55] 29 free
- 1.2. Behrend functions of schemes and stacks, from chapter 4 411
- 1.3. Summary of the main results in chapter 5 512
- 1.4. Examples and applications in chapter 6 714
- 1.5. Extension to quivers with superpotentials in chapter 7 815
- 1.6. Relation to the work of Kontsevich and Soibelman [63] 916

- Chapter 2. Constructible functions and stack functions 1320
- Chapter 3. Background material from [51-54] 2128
- Chapter 4. Behrend functions and Donaldson–Thomas theory 3138
- Chapter 5. Statements of main results 4552
- 5.1. Local description of the moduli of coherent sheaves 4754
- 5.2. Identities on Behrend functions of moduli stacks 5360
- 5.3. A Lie algebra morphism \𝑡𝑖Ψ:\𝑆𝐹𝑎𝑖(\𝑓𝑀)\𝑟𝑎\𝑡𝑖𝐿(𝑋), and generalized Donaldson–Thomas invariants 𝐷𝑇^{\𝑎𝑙}(𝜏) 5461
- 5.4. Invariants 𝑃𝐼^{\𝑎𝑙,𝑛}(𝜏’) counting stable pairs, and deformation-invariance of the 𝐷𝑇^{\𝑎𝑙}(𝜏) 5865

- Chapter 6. Examples, applications, and generalizations 6370
- 6.1. Computing 𝑃𝐼^{𝛼,𝑛}(𝜏’), 𝐷𝑇^{𝛼}(𝜏) and 𝐽^{𝛼}(𝜏) in examples 6370
- 6.2. Integrality properties of the 𝐷𝑇^{𝛼}(𝜏) 6976
- 6.3. Counting dimension zero sheaves 7178
- 6.4. Counting dimension one sheaves 7380
- 6.5. Why it all has to be so complicated: an example 7784
- 6.6. 𝜇-stability and invariants 𝐷𝑇^{𝛼}(𝜇) 8188
- 6.7. Extension to noncompact Calabi–Yau 3-folds 8289
- 6.8. Configuration operations and extended Donaldson–Thomas invariants 8693

- Chapter 7. Donaldson–Thomas theory for quivers with superpotentials 8996
- 7.1. Introduction to quivers 8996
- 7.2. Quivers with superpotentials, and 3-Calabi–Yau categories 9299
- 7.3. Behrend function identities, Lie algebra morphisms, and Donaldson–Thomas type invariants 96103
- 7.4. Pair invariants for quivers 99106
- 7.5. Computing 𝐷𝑇^{𝐝}_{𝐐,𝐈}(𝜇),𝐃𝐓^{𝐝}_{𝐐,𝐈}(𝜇) in examples 104111
- 7.6. Integrality of 𝐷𝑇^{𝐝}_{𝐐}(𝜇) for generic (𝜇,ℝ,⩽) 111118

- Chapter 8. The proof of Theorem 5.3 119126
- Chapter 9. The proofs of Theorems 5.4 and 5.5 123130
- 9.1. Holomorphic structures on a complex vector bundle 124131
- 9.2. Moduli spaces of analytic vector bundles on 𝑋 127134
- 9.3. Constructing a good local atlas 𝑆 for 𝔐 near [𝔈] 128135
- 9.4. Moduli spaces of algebraic vector bundles on 𝑋 130137
- 9.5. Identifying versal families of holomorphic structures and algebraic vector bundles 131138
- 9.6. Writing the moduli space as 𝐶𝑟𝑖𝑡(𝑓) 133140
- 9.7. The proof of Theorem 5.4 135142
- 9.8. The proof of Theorem 5.5 135142

- Chapter 10. The proof of Theorem 5.11 139146
- Chapter 11. The proof of Theorem 5.14 147154
- Chapter 12. The proofs of Theorems 5.22, 5.23 and 5.25 155162
- 12.1. The moduli scheme of stable pairs ℳ_{𝓈𝓉𝓅}^{𝛼,𝓃}(𝜏’) 155162
- 12.2. Pairs as objects of the derived category 157164
- 12.3. Cotangent complexes and obstruction theories 158165
- 12.4. Deformation theory for pairs 160167
- 12.5. A non-perfect obstruction theory for ℳ_{𝓈𝓉𝓅}^{𝛼,𝓃}(𝜏’)/𝒰 163170
- 12.6. A perfect obstruction theory when 𝑟𝑎𝑛𝑘𝛼≠1 167174
- 12.7. An alternative construction for all 𝑟𝑎𝑛𝑘𝛼 170177
- 12.8. Deformation-invariance of the 𝑃𝐼^{𝛼,𝑛}(𝜏’) 173180

- Chapter 13. The proof of Theorem 5.27 175182
- Bibliography 187194
- Glossary of Notation 193200
- Index 197204 free