June 4, 2019
TITLE: Homological mirror symmetry for higher dimensional pants
ABSTRACT: Any Riemann surface can be glued together from pairs-of-pants. This provides a way of proving homological mirror symmetry for Riemann surfaces by first constructing a mirror to a single pair-of-pants and then categorically gluing several copies.
A theorem of Mikhalkin says that complex hypersurfaces in CPn admit decompositions into higher dimensional pairs-of-pants. We prove that the wrapped Fukaya category of the complement of (n+2)-generic hyperplanes in CPn (n-dimensional pants) is equivalent to the derived category of the singular affine variety x1x2..xn+1=0. By taking covers, we also get some simple examples of gluing of pairs-of-pants and the corresponding mirror symmetry statements. Our proof is simple but combines ideas from low-dimensional topology (Heegaard-Floer) and noncommutative resolutions of singularities.
This is joint work with A. Polishchuk.