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# memo_has_moved_text();A theory of generalized Donaldson–Thomas invariants

Dominic Joyce, The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, United Kingdom and Yinan Song, The Mathematical Institute, 24-29 St. Giles, Oxford, OX1 3LB, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 217, Number 1020
ISBNs: 978-0-8218-5279-8 (print); 978-0-8218-8752-3 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2011-00630-1
Published electronically: July 18, 2011
Keywords:Donaldson–Thomas invariant, Calabi–Yau 3-fold, coherent sheaf, vector bundle, stability condition, semistable, Gieseker stability, moduli space, Artin stack
MSC: Primary 14N35; Secondary 14J32, 14F05, 14J60, 14D23

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Chapters

• Chapter 1. Introduction
• Chapter 2. Constructible functions and stack functions
• Chapter 3. Background material from [51-54]
• Chapter 4. Behrend functions and Donaldson–Thomas theory
• Chapter 5. Statements of main results
• Chapter 6. Examples, applications, and generalizations
• Chapter 7. Donaldson–Thomas theory for quivers with superpotentials
• Chapter 8. The proof of Theorem 5.3
• Chapter 9. The proofs of Theorems 5.4 and 5.5
• Chapter 10. The proof of Theorem 5.11
• Chapter 11. The proof of Theorem 5.14
• Chapter 12. The proofs of Theorems 5.22, 5.23 and 5.25
• Chapter 13. The proof of Theorem 5.27

### Abstract

Donaldson-Thomas invariants $DT^\alpha(\tau)$ are integers which count' $\tau$-stable coherent sheaves with Chern character $\alpha$ on a Calabi-Yau 3-fold $X$, where $\tau$ denotes Gieseker stability for some ample line bundle on $X$. They are unchanged under deformations of $X$. The conventional definition works only for classes $\alpha$ containing no strictly $\tau$-semistable sheaves. Behrend showed that $DT^\alpha(\tau)$ can be written as a weighted Euler characteristic $\chi\bigl(\mathcal{M}_{\mathrm{st}}^\alpha(\tau), \nu_{\mathcal{M}_{\mathrm{st}}^\alpha(\tau)}\bigr)$ of the stable moduli scheme $\mathcal{M}_{\mathrm{st}}^\alpha(\tau)$ by a constructible function $\nu_{\mathcal{M}_{\mathrm{st}}^\alpha(\tau)}$ we call the Behrend function'.

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 we study the local structure of the moduli stack $\mathfrak{M}$ of coherent sheaves on $X$. We show that an atlas for $\mathfrak{M}$ may be written locally as $\operatorname{Crit}(f)$ for $f:U\rightarrow\mathbb{C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu_{\mathfrak{M}}$. We compute our invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories $Q$ with relations $I$ coming from a superpotential $W$ on $Q$, and connect our ideas with Szendrői's noncommutative 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 Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.