Localizing virtual cycles for Donaldson-Thomas invariants of Calabi-Yau 4-folds

TL;DR

This paper proves the localization of Donaldson-Thomas virtual cycles for Calabi-Yau 4-folds to zero loci of isotropic cosections, enabling refined invariants and vanishing results for hyperk"ahler 4-folds.

Contribution

It introduces a method to localize DT4 virtual cycles using isotropic cosections, extending the Oh-Thomas construction and applying it to hyperk"ahler 4-folds.

Findings

Virtual cycle localizes to zero locus of cosection

Constructs reduced virtual cycle when cycle vanishes

Proves DT4 vanishing results for hyperk"ahler 4-folds

Abstract

Recently Oh-Thomas constructed a virtual cycle $[X]^{\mathrm{vir}}\in A_*(X)$ for a quasi-projective moduli space $X$ of stable sheaves or complexes over a Calabi-Yau 4-fold against which DT4 invariants may be defined as integrals of cohomology classes. In this paper, we prove that the virtual cycle localizes to the zero locus $X(\sigma)$ of an isotropic cosection $\sigma$ of the obstruction sheaf $Ob_X$ of $X$ and construct a localized virtual cycle $[X]^{\mathrm{vir}}_{\mathrm{loc}}\in A_*(X(\sigma))$. This is achieved by further localizing the Oh-Thomas class which localizes Edidin-Graham's square root Euler class of a special orthogonal bundle. When the cosection $\sigma$ is surjective so that the virtual cycle vanishes, we construct a reduced virtual cycle $[X]^{\mathrm{vir}}_{\mathrm{red}}$. As an application, we prove DT4 vanishing results for hyperk\"ahler 4-folds. All these results hold for virtual structure sheaves and K-theoretic DT4 invariants.