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

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 $L_1^p$ to a limiting $L_1^p$ 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 HyperKähler structure, addressing a conjecture of Fukaya in this setting.

A new result is obtained