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.

中文翻译：

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折叠 $K3$ 表面上的反自对偶连接的绝热极限

我们证明了当纤维塌陷时，在椭圆 $K3$ 表面 $M$ 上的一系列 Yang-Mills 连接的收敛结果。特别是，假设 $M$ 是射影的，承认一个截面，并且具有 Kodaira 类型 $I_1$ 和类型 $II$ 的奇异纤维。令 $\Xi_{t_k}$ 是在 $M$ 上的主要 $SU(n)$ bundle 上的 $SU(n)$ 连接序列，它们相对于 Ricci 平面度量序列是反自对偶的折叠$M$的纤维。给定由 $\bar\partial_{\Xi_{t_k}}$ 引起的光谱覆盖的某些非退化假设，我们证明远离有限数量的光纤，曲率 $F_{\Xi_{t_k}}$ 是局部有界 $C^0$，连接沿 $L^p_1$ 中的子序列（和模酉规范变化）收敛到限制 $L^p_1$ 连接 $\Xi_0$，以及 $\Xi_0$ 的限制对任何纤维是 $C^{1, \alpha}$ 规范等效于具有由光谱覆盖序列确定的全纯结构的平面连接。此外，我们将连接 $\Xi_{t_k}$ 与镜像 HyperK\"ahler 结构中的特殊拉格朗日多截面的收敛族联系起来，解决了 Fukaya 在此设置中的猜想。