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

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 I1 and type II. Let Ξtk 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 ¯¯¯∂Ξtk, we show that away from a finite number of fibers, the curvature FΞtk is locally bounded in C0, the connections converge along a subsequence (and modulo unitary gauge change) in Lp1 to a limiting Lp1 connection Ξ0, and the restriction of Ξ0 to any fiber is C1,α gauge equivalent to a flat connection with holomorphic structure determined by the sequence of spectral covers. Additionally, we relate the connections Ξtk to a converging family of special Lagrangian multi-sections in the mirror HyperKähler structure, addressing a conjecture of Fukaya in this setting.