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

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