Sectorial Decompositions of Symmetric Products of Surfaces and Homological Mirror Symmetry

Abstract

Symmetric products of Riemann surfaces play a crucial role in symplectic geometry and low-dimensional topology. Symmetric products of punctured surfaces are Liouville manifolds of interest e.g. for Heegaard Floer theory. We study the symplectic topology of these spaces using Liouville sectorial techniques, along with examples and applications of these decompositions in the context of homological mirror symmetry. More precisely, we show that a sectorial decomposition of a Riemann surface along a union of arcs induces a sectorial decomposition of its second symmetric product and as an application, we give a new geometric proof of Homological Mirror Symmetry for the complex two dimensional pair of pants.