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

TL;DR

This paper studies the behavior of Hermitian-Yang-Mills connections on stable bundles over collapsing elliptically fibered K3 surfaces, showing convergence to flat connections on generic fibers and relating fiber restrictions to limiting connections.

Contribution

It proves the convergence of Hermitian-Yang-Mills connections on stable bundles over collapsing K3 surfaces and characterizes the limit in terms of fiber restrictions.

Findings

Hermitian-Yang-Mills connections converge to flat connections on generic fibers.

The limit connection is uniquely determined by the fiber restriction data.

Provides insight into the geometric structure of bundles over degenerating 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 $A_0$.