# Homological mirror symmetry for log Calabi--Yau surfaces and applications

# Homological mirror symmetry for log Calabi--Yau surfaces and applications

**In-Person and Online Talk **

Joint work with Ailsa Keating. A log Calabi--Yau surface is a smooth projective surface Y together with a normal crossing divisor D such that K_Y+D=0. If D is singular and negative definite then it can be contracted to a cusp singularity, and the mirror of U:=Y \ D in the sense of Strominger--Yau--Zaslow is the Milnor fiber M of a smoothing of the dual cusp. We prove Kontsevich's homological mirror symmetry conjecture in this setting: the derived category of coherent sheaves on U is isomorphic to the (wrapped) Fukaya category of M. As an application we study the monodromy group for M. In particular we describe compactly supported symplectomorphisms of M which are not compositions of generalized Dehn twists in Lagrangian spheres. Based on arxiv preprints 2005.05010 and 2112.06797.