Memoirs of the American Mathematical Society 2012; 199 pp; softcover Volume: 217 ISBN10: 0821852795 ISBN13: 9780821852798 List Price: US$86 Individual Members: US$51.60 Institutional Members: US$68.80 Order Code: MEMO/217/1020
 This book studies generalized DonaldsonThomas 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  Introduction
 Constructible functions and stack functions
 Background material from [51, 52, 53, 54]
 Behrend functions and DonaldsonThomas theory
 Statements of main results
 Examples, applications, and generalizations
 DonaldsonThomas 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
