Vector bundles on Fano threefolds and K3 surfaces

Abstract

Let X be a Fano threefold, and let $$S\subset X$$ be a K3 surface. Any moduli space $${\mathscr {M}}_{S}$$ 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 $${\mathscr {M}}_{S}$$ . We illustrate this with a number of concrete examples.