Semi-rigid stable sheaves: a criterion and examples

TL;DR

This paper introduces a new criterion for semi-rigidity of stable sheaves on smooth varieties, linking it to the absence of decomposable elements in the Yoneda pairing kernel, with applications to line bundles on various manifolds.

Contribution

It defines semi-rigidity for stable sheaves, provides a detection criterion via the Yoneda pairing, and applies it to line bundles on different geometric contexts.

Findings

Semi-rigidity is characterized by the absence of decomposable elements in the kernel of the Yoneda pairing.

The criterion is applied successfully to line bundles on smooth projective varieties.

Applications include line bundles on Lagrangian subvarieties of hyper-Kähler manifolds.

Abstract

Inspired by Mukai's work on K3 surfaces, we introduce and study a notion of semi-rigidity for stable sheaves on smooth polarised varieties, designed to capture the existence of stable deformations of direct sums. We show that semi-rigidity is detected by the absence of decomposable elements in the kernel of the Yoneda pairing. We apply the resulting criterion to line bundles on smooth projective varieties and to line bundles supported on smooth Lagrangian subvarieties of hyper-K\"ahler manifolds.