Homological mirror symmetry of elementary birational cobordisms

Abstract

The derived category of coherent sheaves ${\mathcal{T}}_B$ associated to a birational cobordism which is either a weighted projective space, a stacky Atiyah flip, or a stacky blow-up of a point has a conjectural mirror Fukaya-Seidel category ${\mathcal{T}}_A$. The potential $W$ defining ${\mathcal{T}}_A$ has base ${\mathbb{C}}^{\ast }$ and exhibits a great deal of symmetry. This paper investigates the structure of the Fukaya-Seidel category for the mirror potentials. A proof of homological mirror symmetry ${\mathcal{T}}_A\cong {\mathcal{T}}_B$ for these birational cobordisms is then given.

39 pages, 8 figures. typos removed