Localizing Virtual Structure Sheaves for Almost Perfect Obstruction Theories

TL;DR

This paper extends localization formulas in $K$-theory to almost perfect obstruction theories, enabling refined invariants and wall crossing formulas for complex moduli stacks, with applications to Donaldson-Thomas theory.

Contribution

It generalizes virtual localization and cosection formulas to almost perfect obstruction theories, enhancing the computation of $K$-theoretic invariants for moduli stacks.

Findings

Established $K$-theoretic wall crossing formulas for $bC^*$-actions.

Defined refined $K$-theoretic invariants for moduli stacks.

Extended localization techniques to almost perfect obstruction theories.

Abstract

Almost perfect obstruction theories were introduced in an earlier paper by the authors as the appropriate notion in order to define virtual structure sheaves and $K$-theoretic invariants for many moduli stacks of interest, including $K$-theoretic Donaldson-Thomas invariants of sheaves and complexes on Calabi-Yau threefolds. The construction of virtual structure sheaves is based on the $K$-theory and Gysin maps of sheaf stacks. In this paper, we generalize the virtual torus localization and cosection localization formulas and their combination to the setting of almost perfect obstruction theory. To this end, we further investigate the $K$-theory of sheaf stacks and its functoriality properties. As applications of the localization formulas, we establish a $K$-theoretic wall crossing formula for simple $\mathbb{C}^\ast$-wall crossings and define $K$-theoretic invariants refining the Jiang-Thomas virtual signed Euler characteristics.