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 \(Q\).