An effective restriction theorem via wall-crossing and Mercat's   conjecture
  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. 
Soheyla Feyzbakhsh
math.AG