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Semi-rigid stable sheaves: a criterion and examples

Alessio Bottini, Riccardo Carini

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-Kähler manifolds.

Semi-rigid stable sheaves: a criterion and examples

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-Kähler manifolds.
Paper Structure (5 sections, 23 theorems, 67 equations)

This paper contains 5 sections, 23 theorems, 67 equations.

Table of Contents

  1. Local structure of Mv(X,h) to Mv(X,h)
  2. The commuting scheme
  3. Local structure of Symn Mv(X,h) to Mnv
  4. Semi-rigid stable sheaves
  5. Examples and applications

Key Result

Theorem 1

A stable sheaf $\mathscr F$ is semi-rigid if and only if $\ker(\Upsilon_{\mathscr{F}})\subseteq \mathop{\raisebox{0.25ex}{$\bigwedge$}}\nolimits^2 \operatorname{Ext}^1(\mathscr{F}, \mathscr{F})$ contains no non-zero decomposable element.

Theorems & Definitions (52)

  • Theorem 1
  • Theorem 2
  • Proposition 3
  • Proposition 4
  • Theorem 5
  • Lemma 1.1
  • proof
  • Remark 1.2: Stable locus
  • Remark 1.3: Compatibility with direct sums
  • Remark 1.4: Moduli of representations and quiver varieties
  • ...and 42 more