Cosection Localization and Vanishing for Virtual Fundamental Classes of D-Manifolds

TL;DR

This paper combines derived differential geometry and cosection localization to establish vanishing results for virtual fundamental classes, demonstrating that certain invariants of hyperk"{a}hler fourfolds are zero.

Contribution

It introduces a new approach by integrating Joyce's derived differential geometry with Kiem-Li's cosection localization to analyze virtual classes.

Findings

Virtual fundamental classes vanish for hyperk"{a}hler fourfolds.

The method unifies derived geometry with localization techniques.

Stable pair invariants are shown to be zero in specific cases.

Abstract

We establish cosection localization and vanishing results for virtual fundamental classes of derived manifolds, combining the theory of derived differential geometry by Joyce with the theory of cosection localization by Kiem-Li. As an application, we show that the stable pair invariants of hyperk\"{a}hler fourfolds, defined by Cao-Maulik-Toda, are zero.