Hermitian–Yang–Mills Connections on Collapsing Elliptically Fibered K3 Surfaces

Let $$X\rightarrow {{\mathbb {P}}}^1$$ X → P 1 be an elliptically fibered K3 surface, admitting a sequence $$\omega _{i}$$ ω i of Ricci-flat metrics collapsing the fibers. Let V be a holomorphic SU(n) bundle over X, stable with respect to $$\omega _i$$ ω i . Given the corresponding sequence $$\Xi _i$$ Ξ i of Hermitian–Yang–Mills connections on V, we prove that, if E is a generic fiber, the restricted sequence $$\Xi _i|_{E}$$ Ξ i | E converges to a flat connection $$A_0$$ A 0 . Furthermore, if the restriction $$V|_E$$ V | E is of the form $$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$ ⊕ j = 1 n O E ( q j - 0 ) for n distinct points $$q_j\in E$$ q j ∈ E , then these points uniquely determine $$A_0$$ A 0 .