Homological mirror symmetry for log Calabi–Yau surfaces

Hacking, Paul¹ ; Keating, Ailsa²

¹ Department of Mathematics and Statistics, University of Massachusetts, Amherst, Amherst, MA, United States
² Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, University of Cambridge, Cambridge, United Kingdom

Geometry & topology, Tome 26 (2022) no. 8, pp. 3747-3833.

Résumé

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 $D^b\mathrm{Coh}(Y)$, and the wrapped Fukaya category $D^b\mathcal{𝒲}(M)$ is isomorphic to $D^b\mathrm{Coh}(Y\setminus D)$. 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).

Détail

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

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