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

Abstract

AbstractWe prove an effective restriction theorem for stable vector bundles E on a smooth projective variety: $$E|_D$$ E | D is (semi)stable for all irreducible divisors $$D \in |kH|$$ D ∈ | k H | 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 $${\mathbb {P}}^n$$ P n restricted to a K3 surface.