Homological Mirror Symmetry and Simple Elliptic Singularities

We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic hypersurface singularities.Comment: 17 pages. v2: note added at the end of Introductio

Homological Mirror Symmetry for Toric del Pezzo Surfaces

We prove the homological mirror conjecture for toric del Pezzo surfaces. In this case, the mirror object is a regular function on an algebraic torus. We show that the derived Fukaya category of this mirror coincides with the derived category of coherent sheaves on the original manifold.Comment: 19 pages, 6 figure

A Remark on a Theorem of math.AG/0511155

We give another proof of a theorem of H. Kajiura, K. Saito, and A. Takahashi based on the theory of weighted projective lines by Geigle and Lenzing and a theorem of Orlov on triangulated categories of graded B-branes. The content of this paper appears in the appendix to math.AG/0511155.Comment: The content of this paper appears in the appendix to math.AG/051115

Triangulated categories of Gorenstein cyclic quotient singularities

We prove an equivalence of triangulated categories between Orlov's triangulated category of singularities for a Gorenstein cyclic quotient singularity and the derived category of representations of a quiver with relations which is obtained from the McKay quiver by removing one vertex and half of the arrows.Comment: 3 pages, no figures; v2: removed an error pointed out by Kentaro Naga

Mirror symmetry and K3 surfaces

We review some of the interplay between mirror symmetry and K3 surfaces.Comment: 37 pages, 3 figures; v2: references adde

Stokes Matrices for the Quantum Cohomologies of Grassmannians

We prove the conjectural relation between the Stokes matrix for the quantum cohomology and an exceptional collection generating the derived category of coherent sheaves in the case of the Grassmannian. The proof is based on the relation between the quantum cohomology of the Grassmannian and that of the projective space.Comment: 11 pages, 1 figur

Hyperplane sections and stable derived categories

We discuss the relation between the graded stable derived category of a hypersurface and that of its hyperplane section. The motivation comes from the compatibility between homological mirror symmetry for the Calabi-Yau manifold defined by an invertible polynomial and that for the singularity defined by the same polynomial.Comment: 11 pages, v2: removed an error pointed out by Yukinobu Tod

Goldman systems and bending systems

We show that the moduli space of parabolic bundles on the projective line and the polygon space are isomorphic, both as complex manifolds and symplectic manifolds equipped with structures of completely integrable systems, if the stability parameters are small.Comment: 37 pages, 2 figures; v3: added a referenc

Quantum entanglement, Calabi-Yau manifolds, and noncommutative algebraic geometry

We relate SLOCC equivalence classes of qudit states to moduli spaces of Calabi-Yau manifolds equipped with a collection of line bundles. The cases of 3 qutrits and 4 qubits are also related to noncommutative algebraic geometry.Comment: 11 pages. Comments welcome

Potential functions on Grassmannians of planes and cluster transformations

With a triangulation of a planar polygon with $n$ sides, one can associate an integrable system on the Grassmannian of 2-planes in an $n$-space. In this paper, we show that the potential functions of Lagrangian torus fibers of the integrable systems associated with different triangulations glue together by cluster transformations. We also prove that the cluster transformations coincide with the wall-crossing formula in Lagrangian intersection Floer theory.Comment: 47 pages. v2: revised following referee's comment