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A Theory of Generalized Donaldson-Thomas Invariants
Dominic Joyce, The Mathematical Institute, Oxford, United Kingdom, and Yinan Song, Budapest, Hungary
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Memoirs of the American Mathematical Society
2012; 199 pp; softcover
Volume: 217
ISBN-10: 0-8218-5279-5
ISBN-13: 978-0-8218-5279-8
List Price: US$86 Individual Members: US$51.60
Institutional Members: US\$68.80
Order Code: MEMO/217/1020

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 $$Q$$.

• Introduction
• Constructible functions and stack functions
• Background material from [51, 52, 53, 54]
• Behrend functions and Donaldson-Thomas theory
• Statements of main results
• Examples, applications, and generalizations
• Donaldson-Thomas theory for quivers with superpotentials
• The proof of Theorem 5.3
• The proofs of Theorems 5.4 and 5.5
• The proof of Theorem 5.11
• The proof of Theorem 5.14
• The proofs of Theorems 5.22, 5.23 and 5.25
• The proof of Theorem 5.27
• Bibliography
• Glossary of Notation
• Index