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Homological mirror symmetry for log Calabi–Yau surfaces

Paul Hacking and Ailsa Keating

Appendix: Wendelin Lutz

Geometry & Topology 26 (2022) 3747–3833
DOI: 10.2140/gt.2022.26.3747

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 , where M is a Weinstein four-manifold, such that the directed Fukaya category of w is isomorphic to Db Coh (Y ), and the wrapped Fukaya category Db𝒲(M) is isomorphic to Db 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).

cusp singularities, homological mirror symmetry, Fukaya categories, coherent sheaves, Lefschetz fibrations
Mathematical Subject Classification
Primary: 53D37
Secondary: 14B05, 18G70
Received: 18 February 2021
Revised: 4 August 2021
Accepted: 4 September 2021
Published: 16 March 2023
Proposed: Paul Seidel
Seconded: Mark Gross, Yakov Eliashberg
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
United Kingdom
Wendelin Lutz
Department of Mathematics
Imperial College London
United Kingdom