Homological mirror symmetry for log Calabi-Yau surfaces

2020/05/11 by Hacking, Paul, Keating, Ailsa, Lutz, Wendelin · 2 citations
#53D37 (Primary) 14B05 #53D40 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2005.05010

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 → 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^bW (M) is isomorphic to D^b 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.

Cited by

• Lectures on stabilized ellipsoid embeddings
• On the mirrors of low-degree del Pezzo surfaces

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