二木 昌宏

We introduce the notion of a tropical coamoeba which gives a combinatorial description of the Fukaya category of the mirror of a toric Fano stack. We show that the polyhedral decomposition of a real n-torus into n + 1 permutohedra gives a tropical coamoeba for the mirror of the projective space , and we prove a torus-equivariant version of homological mirror symmetry for the projective space. As a corollary, we obtain homological mirror symmetry for toric orbifolds of the projective space.

We prove homological mirror symmetry for Lefschetz fibrations obtained as Sebastiani-Thom sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under Sebastiani-Thom summation of a polynomial of type D. © 2012 Springer-Verlag.

We prove that the derived Fukaya category of the Lefschetz fibration defined by a Brieskorn-Pham polynomial is equivalent to the triangulated category of singularities associated with the same polynomial together with a grading by an abelian group of rank one. Symplectic Picard-Lefschetz theory developed by Seidel is an essential ingredient of the proof.

We associate an exact Lefshetz fibration with a pair of a consistent dimer model and an internal perfect matching on it, whose Fukaya category is derived equivalent to the category of representations of the directed quiver with relations associated with the pair. As a corollary, we obtain a version of homological mirror symmetry for two-dimentional toric Fano Stacks.