A theory of generalized Donaldson-Thomas invariants

Donaldson-Thomas invariants $DT^\alpha(\tau)$ are integers which `count' $\tau$-stable coherent sheaves with Chern character $\alpha$ on a Calabi-Yau 3-fold $X$, where $\tau$ denotes Gieseker stability for some ample line bundle on $X$. They are unchanged under deformations of $X$. The conventional definition works only for classes $\alpha$ containing no strictly $\tau$-semistable sheaves. Behrend showed that $DT^\alpha(\tau)$ can be written as a weighted Euler characteristic $\chi\bigl(\M_{\mathrm{st}}^\alpha(\tau), \nu_{\M_{\mathrm{st}}^\alpha(\tau)}\bigr)$ of the stable moduli scheme $\M_{\mathrm{st}}^\alpha(\tau)$ by a constructible function $\nu_{\M_{\mathrm{st}}^\alpha(\tau)}$ we call the `Behrend function'.

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 we study the local structure of the moduli stack $\fM$ of coherent sheaves on $X$. We show that an atlas for $\fM$ may be written locally as $\Crit(f)$ for $f:U\ra\C$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $\nu_\fM$. We compute our invariants $\bar{DT}{}^\alpha(\tau)$ in examples, and make a conjecture about their integrality properties. We also extend the theory to abelian categories $\modCQI$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$, and connect our ideas with Szendrői's noncommutative Donaldson-Thomas invariants, and work by Reineke and others on invariants counting quiver representations. Our book is closely related to Kontsevich and Soibelman's independent paper Stability structures, motivic Donaldson-Thomas invariants and cluster transformations.