Hermitian-Yang-Mills connections on collapsing elliptically fibered $K3$ surfaces

Abstract

Let $X\to {\mathbb{P}}^1$ 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 ${\oplus }_{j=1}^n{\mathcal{O}}_E(q_j-0)$ for $n$ distinct points $q_j\in E$, then these points uniquely determine $A_0$.

arXiv commentThis version contains new notation, further expository details, and a breaking of the main result into a Theorem and Corollary, as suggested by the referee