Cosection Localization for D-Manifolds and $(-2)$-Shifted Symplectic Derived Schemes, Revisited

TL;DR

This paper advances the theory of cosection localization for d-manifolds and $(-2)$-shifted symplectic derived schemes, establishing new constructions and demonstrating their equivalence in homology, with applications to Donaldson-Thomas invariants.

Contribution

It constructs reduced and cosection localized virtual fundamental classes in broader contexts and proves their equivalence using recent algebraic and differential geometric results.

Findings

Construction of reduced virtual fundamental classes for derived manifolds.

Cosection localized virtual classes for $(-2)$-shifted symplectic schemes in greater generality.

Equivalence of algebraic and differential geometric constructions in homology.

Abstract

This is a continuation of prior work of the author on cosection localization for d-manifolds. We construct reduced virtual fundamental classes for derived manifolds with surjective cosections and cosection localized virtual fundamental classes for $(-2)$-shifted symplectic derived schemes in larger generality. Moreover, using recent results of Oh-Thomas, we show that the algebraic and differential geometric constructions of reduced and cosection localized virtual fundamental classes of $(-2)$-shifted symplectic derived schemes yield the same result in homology. We obtain applications towards the construction and integrality of reduced invariants in Donaldson-Thomas theory of Calabi-Yau fourfolds.