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A Theory of Generalized Donaldson-Thomas Invariants

Dominic Joyce The Mathematical Institute, Oxford, United Kingdom
Yinan Song , Budapest, Hungary
Available Formats:
Electronic ISBN: 978-0-8218-8752-3
Product Code: MEMO/217/1020.E
List Price: $86.00 MAA Member Price:$77.40
AMS Member Price: $51.60 Click above image for expanded view A Theory of Generalized Donaldson-Thomas Invariants Dominic Joyce The Mathematical Institute, Oxford, United Kingdom Yinan Song , Budapest, Hungary Available Formats:  Electronic ISBN: 978-0-8218-8752-3 Product Code: MEMO/217/1020.E  List Price:$86.00 MAA Member Price: $77.40 AMS Member Price:$51.60
• Book Details

Memoirs of the American Mathematical Society
Volume: 2172012; 199 pp
MSC: Primary 14;

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 $Q$.

• 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
• Requests

Review Copy – for reviewers who would like to review an AMS book
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Volume: 2172012; 199 pp
MSC: Primary 14;

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 $Q$.

• 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
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