Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces

TL;DR

This paper proves a convergence result for anti-self-dual Yang-Mills connections on collapsing elliptic K3 surfaces, relating the limit to flat connections and special Lagrangian multi-sections, thus addressing a conjecture of Fukaya.

Contribution

It establishes the convergence of anti-self-dual connections on collapsing K3 surfaces and links the limit to flat connections and mirror symmetry structures.

Findings

Curvature remains bounded away from singular fibers.

Connections converge in L^p_1 to flat connections on fibers.

Relation between gauge-theoretic limits and special Lagrangian multi-sections.

Abstract

We prove a convergence result for a family of Yang-Mills connections over an elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is projective, admits a section, and has singular fibers of Kodaira type $I_1$ and type $II$. Let $\Xi_{t_k}$ be a sequence of $SU(n)$ connections on a principal $SU(n)$ bundle over $M$, that are anti-self-dual with respect to a sequence of Ricci flat metrics collapsing the fibers of $M$. Given certain non-degeneracy assumptions on the spectral covers induced by $\bar\partial_{\Xi_{t_k}}$, we show that away from a finite number of fibers, the curvature $F_{\Xi_{t_k}}$ is locally bounded in $C^0$, the connections converge along a subsequence (and modulo unitary gauge change) in $L^p_1$ to a limiting $L^p_1$ connection $\Xi_0$, and the restriction of $\Xi_0$ to any fiber is $C^{1,\alpha}$ gauge equivalent to a flat connection with holomorphic structure determined by the sequence of spectral covers. Additionally, we relate the connections $\Xi_{t_k}$ to a converging family of special Lagrangian multi-sections in the mirror HyperK\"ahler structure, addressing a conjecture of Fukaya in this setting.