Hyperbolic localization of the Donaldson-Thomas sheaf

TL;DR

This paper establishes a toric localization formula in cohomological Donaldson-Thomas theory for -1-shifted symplectic spaces with C* actions, generalizing Bialynicki-Birula decomposition to a derived setting.

Contribution

It introduces a -1-shifted version of the Bialynicki-Birula decomposition for Donaldson-Thomas sheaves under C* actions on symplectic spaces.

Findings

Expresses DT sheaf restriction as a sum over fixed components.

Generalizes classical localization to derived symplectic geometry.

Applicable to moduli spaces of sheaves on Calabi-Yau threefolds.

Abstract

In this paper we prove a toric localization formula in cohomological Donaldson Thomas theory. Consider a -1-shifted symplectic algebraic space with a C* action leaving the -1-shifted symplectic form invariant. This includes the moduli space of stable sheaves or complexes of sheaves on a Calabi-Yau threefold with a C*-invariant Calabi-Yau form, or the intersection of two C*-invariant Lagrangians in a symplectic space with a C*-invariant symplectic form. In this case we express the restriction of the Donaldson-Thomas perverse sheaf (or monodromic mixed Hodge module) defined by Joyce et al. to the attracting variety as a sum of cohomological shifts of the DT perverse sheaves on the C* fixed components. This result can be seen as a -1-shifted version of the Bialynicki-Birula decomposition for smooth schemes.