A theory of generalized Donaldson-Thomas invariants

TL;DR

This paper introduces a generalized framework for Donaldson-Thomas invariants on Calabi-Yau 3-folds, extending their definition to all Chern characters, establishing deformation invariance, and analyzing their behavior under stability condition changes.

Contribution

It defines new rational invariants a(t) that generalize classical DT invariants, applicable to all Chern characters, and studies their properties and transformations.

Findings

Invariants are deformation-invariant and rational for all classes.

Local structure of moduli stack is described via critical loci of holomorphic functions.

Conjecture on integrality properties of the new invariants.

Abstract

Let X be a Calabi-Yau 3-fold over C. The Donaldson-Thomas invariants of X are integers DT^a(t) which count stable sheaves with Chern character a on X, with respect to a Gieseker stability condition t. They are defined only for Chern characters a for which there are no strictly semistable sheaves on X. They have the good property that they are unchanged under deformations of X. Their behaviour under change of stability condition t was not understood until now. This book defines and studies a generalization of Donaldson-Thomas invariants. Our new invariants \bar{DT}^a(t) are rational numbers, defined for all Chern characters a, and are equal to DT^a(t) if there are no strictly semistable sheaves in class a. They are deformation-invariant, and have a known transformation law under change of stability condition. To prove all this we study the local structure of the moduli stack M of coherent sheaves on X. We show that an atlas for M may be written locally as Crit(f) for f a holomorphic function on a complex manifold, and use this to deduce identities on the Behrend function of M. We compute our invariants in examples, and make a conjecture about their integrality properties. We extend the theory to abelian categories of representations of a quiver with relations coming from a superpotential, and connect our ideas with Szendroi's "noncommutative Donaldson-Thomas invariants" and work by Reineke and others. This book is surveyed in the paper arXiv:0910.0105.