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 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 P n , and prove a torusequivariant 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 disconnected sums of polynomials of types A or D. The proof is based on the behavior of the Fukaya category under the addition of a polynomial of type D.

We associate an exact Lefschetz fibration with a pair of a consistent dimer model and an internal perfect matching on it, whose Fukaya category is derivedequivalent 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-dimensional toric Fano stacks.

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This is a sequel to our paper [11], where we proposed a definition of the Morse homotopy of the moment polytope of toric manifolds. Using this as the substitute of the Fukaya category of the toric manifolds, we proved a version of homological mirror symmetry for the projective spaces and their products via Strominger-Yau-Zaslow construction of the mirror dual Landau-Ginzburg model.In this paper we go this way further and extend our previous result to the case of the Hirzebruch surface F 1 .

In this paper we define an equivariant Floer A∞ algebra for C and CP 1 by using Cartan model. We then prove an equivariant homological mirror symmetry, i.e. an equivalence between an A∞ category of equivariant Lagrangian branes and the category of matrix factorizations of Givental's equivariant Landau-Ginzburg potential function.