eBook ISBN:  9780821887523 
Product Code:  MEMO/217/1020.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $51.60 
eBook ISBN:  9780821887523 
Product Code:  MEMO/217/1020.E 
List Price:  $86.00 
MAA Member Price:  $77.40 
AMS Member Price:  $51.60 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 217; 2012; 199 ppMSC: 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 wallcrossing 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

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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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 wallcrossing 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