Homological mirror symmetry for log Calabi-Yau surfaces · arXivDesk
2005.05010May 11, 2020This is the final version before publication, incorporating the appendix by Lutz, previously missing from the arxiv. Main article by Hacking and Keating. Comments welcome!
Homological mirror symmetry for log Calabi-Yau surfaces
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
Nearby in the stack
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 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.