Vector bundles on Fano threefolds and K3 surfaces

Abstract

Let X be a Fano threefold, and let S be a K3 surface in X . Any moduli space M of simple vector bundles on S carries a holomorphic symplectic structure. Following an idea of Tyurin, we show that in some cases, those vector bundles which come from X form a Lagrangian subvariety of M . We illustrate this with a number of concrete examples.