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

Abstract

Let $X\rightarrow {\mathbb P}^1$ 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 $A_0$.