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June 2021 Adiabatic limits of anti-self-dual connections on collapsed $K3$ surfaces
Ved Datar, Adam Jacob, Yuguang Zhang
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J. Differential Geom. 118(2): 223-296 (June 2021). DOI: 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 $I_1$ 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 $\overline{\partial}_{\Xi_{t_k}}$, 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 HyperKähler structure, addressing a conjecture of Fukaya in this setting.


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Ved Datar. Adam Jacob. Yuguang Zhang. "Adiabatic limits of anti-self-dual connections on collapsed $K3$ surfaces." J. Differential Geom. 118 (2) 223 - 296, June 2021.


Received: 29 September 2018; Published: June 2021
First available in Project Euclid: 3 June 2021

Digital Object Identifier: 10.4310/jdg/1622743140

Rights: Copyright © 2021 Lehigh University


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Vol.118 • No. 2 • June 2021
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