English

Hermitian-Yang-Mills connections on collapsing elliptically fibered $K3$ surfaces

Differential Geometry 2022-08-11 v2

Abstract

Let XP1X\rightarrow {\mathbb P}^1 be an elliptically fibered K3K3 surface, admitting a sequence ωi\omega_{i} of Ricci-flat metrics collapsing the fibers. Let VV be a holomorphic SU(n)SU(n) bundle over XX, stable with respect to ωi\omega_i. Given the corresponding sequence Ξi\Xi_i of Hermitian-Yang-Mills connections on VV, we prove that, if EE is a generic fiber, the restricted sequence ΞiE\Xi_i|_{E} converges to a flat connection A0A_0. Furthermore, if the restriction VEV|_E is of the form j=1nOE(qj0)\oplus_{j=1}^n\mathcal O_E(q_j-0) for nn distinct points qjEq_j\in E, then these points uniquely determine A0A_0.

Keywords

Cite

@article{arxiv.1710.03898,
  title  = {Hermitian-Yang-Mills connections on collapsing elliptically fibered $K3$ surfaces},
  author = {Ved Datar and Adam Jacob},
  journal= {arXiv preprint arXiv:1710.03898},
  year   = {2022}
}

Comments

This version contains new notation, further expository details, and a breaking of the main result into a Theorem and Corollary, as suggested by the referee