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

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.