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

Datar, V and Jacob, A and Zhang, Y (2021) Adiabatic limits of anti-self-dual connections on collapsed K3 surfaces. In: Journal of Differential Geometry, 118 (2). pp. 223-296.

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Official URL: https://doi.org/10.4310/jdg/1622743140


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 �tk 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 �¯�tk, we show that away from a finite number of fibers, the curvature F�tk is locally bounded in C0, the connections converge along a subsequence (and modulo unitary gauge change) in Lp1 to a limiting Lp1 connection �0, and the restriction of �0 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 �tk to a converging family of special Lagrangian multi-sections in the mirror HyperKähler structure, addressing a conjecture of Fukaya in this setting. © 2021 International Press of Boston, Inc.. All rights reserved.

Item Type: Journal Article
Publication: Journal of Differential Geometry
Publisher: International Press, Inc.
Additional Information: The copyright for this article belongs to International Press.
Department/Centre: Division of Physical & Mathematical Sciences > Mathematics
Date Deposited: 02 Aug 2021 08:53
Last Modified: 02 Aug 2021 08:53
URI: http://eprints.iisc.ac.in/id/eprint/69008

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