K-Theoretic Generalized Donaldson-Thomas Invariants

TL;DR

This paper develops a new framework for defining and analyzing K-theoretic Donaldson-Thomas invariants on Calabi-Yau threefolds using almost perfect obstruction theories, ensuring deformation invariance of the associated virtual structure sheaves.

Contribution

It introduces almost perfect obstruction theories on stacks, enabling the definition of K-theoretic Donaldson-Thomas invariants with deformation invariance.

Findings

Established the existence of almost perfect obstruction theories for many stacks.

Constructed a K-theoretic Gysin map for sheaf stacks.

Defined virtual structure sheaves and generalized DT invariants for Calabi-Yau threefolds.

Abstract

We introduce the notion of almost perfect obstruction theory on a Deligne-Mumford stack and show that stacks with almost perfect obstruction theories have virtual structure sheaves which are deformation invariant. The main components in the construction are an induced embedding of the coarse moduli sheaf of the intrinsic normal cone into the associated obstruction sheaf stack and the construction of a $K$-theoretic Gysin map for sheaf stacks. We show that many stacks of interest admit almost perfect obstruction theories. As a result, we are able to define virtual structure sheaves and $K$-theoretic classical and generalized Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds.