An effective restriction theorem via wall-crossing and Mercat's conjecture

TL;DR

This paper establishes an effective restriction theorem for stable vector bundles on smooth projective varieties using wall-crossing techniques, and applies it to disprove Mercat's conjecture for certain curves on K3 surfaces.

Contribution

It introduces a new effective restriction theorem for stable bundles via wall-crossing and applies it to refute Mercat's conjecture in higher ranks on K3 surfaces.

Findings

Proves stability of restrictions for all large divisors.

Disproves Mercat's conjecture for curves on K3 surfaces in ranks > 2.

Reproves stability of tangent bundles on projective spaces.

Abstract

We prove an effective restriction theorem for stable vector bundles $E$ on a smooth projective variety: $E|_D$ is (semi)stable for all irreducible divisors $D \in |kH|$ for all $k$ greater than an explicit constant. As an application, we show that Mercat's conjecture in any rank greater than $2$ fails for curves lying on K3 surfaces. Our technique is to use wall-crossing with respect to (weak) Bridgeland stability conditions which we also use to reprove Camere's result on slope stability of the tangent bundle of $\mathbb{P}^n$ restricted to a K3 surface.