Homological mirror symmetry for log Calabi-Yau surfaces

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.

This is the final version before publication, incorporating the appendix by Lutz, previously missing from the arxiv. Main article by Hacking and Keating. Comments welcome!

References in corpus (4)

• Géométrie de contact: de la dimension trois vers les dimensions supérieures
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Arithmetic mirror symmetry for the 2-torus
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