AMS Bookstore LOGO amslogo
Return to List

AMS TextbooksAMS Applications-related Books

A Theory of Generalized Donaldson-Thomas Invariants
Dominic Joyce, The Mathematical Institute, Oxford, United Kingdom, and Yinan Song, Budapest, Hungary
cover
SEARCH THIS BOOK:

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
[Add Item]

Request Permissions

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 \(Q\).

Table of Contents

  • 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
Powered by MathJax

  AMS Home | Comments: webmaster@ams.org
© Copyright 2014, American Mathematical Society
Privacy Statement

AMS Social

AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia