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A theory of generalized Donaldson–Thomas invariants

About this Title

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

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: https://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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Table of Contents

Chapters

• 1. Introduction
• 2. Constructible functions and stack functions
• 3. Background material
• 4. Behrend functions and Donaldson–Thomas theory
• 5. Statements of main results
• 6. Examples, applications, and generalizations
• 7. Donaldson–Thomas theory for quivers with superpotentials
• 8. The proof of Theorem
• 9. The proofs of Theorems and
• 10. The proof of Theorem
• 11. The proof of Theorem
• 12. The proofs of Theorems , and
• 13. The proof of Theorem

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 $\textrm {Crit}(f)$ 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 mod-$\mathbb {C}Q/I$ of representations of a quiver $Q$ 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.