Limit stable objects on Calabi-Yau 3-folds

TL;DR

This paper introduces new curve counting invariants on Calabi-Yau 3-folds via limit stability in the derived category, generalizing stable pair invariants and exploring wall-crossing phenomena.

Contribution

It develops the notion of limit stability on perverse coherent sheaves and constructs moduli spaces for these objects, leading to new enumerative invariants.

Findings

Defined limit stable objects and their moduli spaces

Constructed new counting invariants generalizing stable pairs

Analyzed wall-crossing behavior of the invariants

Abstract

In this paper, we introduce new enumerative invariants of curves on Calabi-Yau 3-folds via certain stable objects in the derived category of coherent sheaves. We introduce the notion of limit stability on the category of perverse coherent sheaves, a subcategory in the derived category, and construct the moduli spaces of limit stable objects. We then define the counting invariants of limit stable objects using Behrend's constructible functions on that moduli spaces. It will turn out that our invariants are generalizations of counting invariants of stable pairs introduced by Pandharipande and Thomas. We will also investigate the wall-crossing phenomena of our invariants under change of stability conditions.