Homological mirror symmetry for log Calabi–Yau surfaces

Given a log Calabi–Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w : M \to ℂ$ , where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^{b} Coh (Y)$ , and the wrapped Fukaya category $D^{b} 𝒲 (M)$ is isomorphic to $D^{b} Coh (Y ∖ D)$ . We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in work of Gross, Hacking and Keel (Publ. Math. Inst. Hautes Études Sci. 122 (2015) 65–168); when $D$ is negative definite this is expected to be the Milnor fibre of a smoothing of the dual cusp of $D “$ . We also match our mirror potential $w$ with existing constructions for a range of special cases of $(Y, D)$ , notably those of Auroux, Katzarkov and Orlov (Invent. Math. 166 (2006) 537–582) and Abouzaid (Selecta Math. 15 (2009) 189–270).

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Keywords

cusp singularities, homological mirror symmetry, Fukaya categories, coherent sheaves, Lefschetz fibrations

Mathematical Subject Classification

Primary: 53D37

Secondary: 14B05, 18G70

References

Publication

Received: 18 February 2021

Revised: 4 August 2021

Accepted: 4 September 2021

Published: 16 March 2023

Proposed: Paul Seidel

Seconded: Mark Gross, Yakov Eliashberg

Authors

Paul Hacking
Department of Mathematics and Statistics University of Massachusetts, Amherst Amherst, MA United States

Ailsa Keating
Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Cambridge United Kingdom

Wendelin Lutz
Department of Mathematics Imperial College London London United Kingdom

Volume 26, issue 8 (2022)

Paul Hacking and Ailsa Keating

Appendix: Wendelin Lutz

Abstract