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arXiv · 2005.05010

Homological mirror symmetry for log Calabi-Yau surfaces

Paul Hacking·Ailsa Keating·Wendelin Lutz
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Abstract

Given a log Calabi-Yau surface $Y$ with maximal boundary $D$ and distinguished complex structure, we explain how to construct a mirror Lefschetz fibration $w: M \to \mathbb{C}$, where $M$ is a Weinstein four-manifold, such that the directed Fukaya category of $w$ is isomorphic to $D^b \text{Coh}(Y)$, and the wrapped Fukaya category $D^b\mathcal{W} (M)$ is isomorphic to $D^b \text{Coh}(Y \backslash D)$. We construct an explicit isomorphism between $M$ and the total space of the almost-toric fibration arising in the work of Gross-Hacking-Keel; 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 in work of Auroux-Katzarkov-Orlov and Abouzaid.

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Paul Hacking, Ailsa Keating, Wendelin Lutz. 2020-05-11. Homological mirror symmetry for log Calabi-Yau surfaces. https://doi.org/10.2140/gt.2022.26.3747

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