Stable pairs of 2-dimensional sheaves on 4-folds

TL;DR

This paper extends the concept of stable pairs from 3-folds to 4-folds, establishing moduli space correspondences, categorical frameworks, and new invariants for counting 2-dimensional sheaves and pairs on complex four-dimensional varieties.

Contribution

It introduces a new framework for stable pairs on 4-folds, generalizes existing theories, and constructs novel invariants related to 2-dimensional sheaves and pairs, including their categorical and Hall algebra aspects.

Findings

Identified moduli spaces of limit stable pairs with derived category complexes.

Established categorical correspondences involving stable pairs, ideal sheaves, and 1-dimensional sheaves.

Constructed new invariants for Calabi-Yau 4-folds and computed examples for fibrations and local geometries.

Abstract

We identify Le Potier's moduli spaces of limit stable pairs $(F,s)$, where $F$ is a 2-dimensional sheaf on a nonsingular projective 4-fold $X$ and $s \in H^0(F)$, with the moduli spaces of polynomial stable 2-term complexes in derived category. These stable pairs are 2-dimensional analogs of Pandharipande-Thomas' stable pairs defined for 3-folds. We establish categorical correspondences involving these stable pairs, ideal sheaves of 2-dimensional subschemes of $X$, and 1-dimensional sheaves on $X$. Under some conditions on the Chern character, these lead to Hall algebra correspondences. The generalization of most of these results to higher ranks is also given. In case $X$ is Calabi-Yau, Oh-Thomas' construction gives a new set of invariants of $X$ counting these stable pairs. For certain Chern characters, these are related to the invariants of 2-dimensional stable sheaves. We calculate and study them in some cases and examples such as fibrations by abelian surfaces, local surfaces, and local Fano 3-folds. The last case in particular leads to new invariants of Fano 3-folds counting 2-dimensional stable pairs with reduced supports.