Homological mirror symmetry for log Calabi-Yau surfaces

Given a log Calabi-Yau surface Y with maximal boundary D and distinguished com-plex 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 DbCoh(Y ), and the wrapped Fukaya category DbW(M) is isomorphic to DbCoh(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 etudes 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).