Hermitian-Yang-Mills Connections on Collapsing Elliptically Fibered K3 Surfaces.

Abstract

Let $X\to P^1$ be an elliptically fibered K3 surface, admitting a sequence ${\omega }_i$ of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to ${\omega }_i$ . Given the corresponding sequence ${\Xi }_i$ of Hermitian-Yang-Mills connections on V, we prove that, if E is a generic fiber, the restricted sequence ${\Xi }_i{|}_E$ converges to a flat connection $A_0$ . Furthermore, if the restriction ${V|}_E$ is of the form ${\oplus }_{j=1}^n{O}_E(q_j-0)$ for n distinct points $q_j\in E$ , then these points uniquely determine $A_0$ .