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

Ved Datar, Adam Jacob, Yuguang Zhang

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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.
Original languageEnglish
Pages (from-to)223-296
Number of pages74
JournalJournal of Differential Geometry
Issue number2
Early online date3 Jun 2021
Publication statusPublished - 30 Jun 2021

Bibliographical note

Funding Information:
∗ Supported in part by NSF RTG grant DMS-1344991. † Supported in part by a grant from the Hellman Foundation. ‡ Supported by the Simons Foundation under the Simons Collaboration on Special 43 44 Holonomy in Geometry, Analysis and Physics (grant #488620). Received September 29, 2018.

Publisher Copyright:
© 2021 International Press of Boston, Inc.. All rights reserved.

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology


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