These are the notes of two lectures given at the Mini Spring School An introduction to the mathematics of string theory held at Adelaide University in November 2002. It is a leisurely introduction to the mathematics surrounding toric varieties. Lecture I. Aims of Lecture I. (i) To contrast topological, Riemannian, symplectic and complex structures; (ii) to set up various topological objects tha...

We consider subtorus actions on divisorial toric varieties. Here divisoriality means that the variety has many Cartier divisors like quasiprojective and smooth ones. We characterize when a subtorus action on such a toric variety admits a categorical quotient in the category of divisorial varieties. Our result generalizes previous statements for the quasiprojective case. A first step in the proo...

We apply ideas from intersection theory on toric varieties to tropical intersection theory. We introduce mixed Minkowski weights on toric varieties which interpolate between equivariant and ordinary Chow cohomology classes on complete toric varieties. These objects fit into the framework of tropical intersection theory developed by Allermann and Rau. Standard facts about intersection theory on ...

Let (M, σ, ψ) be a symplectic-toric manifold of dimension at least four. This paper investigates the symplectic ball packing problem in the toral equivariant setting. We show that the set of toric symplectic ball packings of M admits the structure of a convex polytope. Previous work of the first author shows that up to equivalence, only (CP)2 and CP admit density one packings when n = 2 and onl...

The toric residue mirror conjecture of Batyrev and Materov [2] for Calabi-Yau hypersurfaces in Gorenstein toric Fano varieties expresses a toric residue as a power series whose coefficients are certain integrals over moduli spaces. This conjecture was proved independently by Szenes and Vergne [10] and Borisov [5]. We build on the work of these authors to generalize the residue mirror map to not...

Toric codes are a class of m-dimensional cyclic codes introduced recently by J. Hansen in [7], [8], and studied in [9], [5], [10]. They may be defined as evaluation codes obtained from monomials corresponding to integer lattice points in an integral convex polytope P ⊆ Rm. As such, they are in a sense a natural extension of Reed-Solomon codes. Several articles cited above use intersection theor...

Toric origami manifold is a generalization of a symplectic toric manifold: it is a manifold with torus action and a compatible 2-form degenerating in a nice, controllable way. In the same way as symplectic toric manifolds are classified by Delzant polytopes, symplectic toric manifolds are classified by certain collections of Delzant polytopes, called origami templates. We are interested in the ...

The optimal density function assigns to each symplectic toric manifold M a number 0 < d ≤ 1 obtained by considering the ratio between the maximum volume of M which can be filled by symplectically embedded disjoint balls and the total symplectic volume of M . In the toric version of this problem, M is toric and the balls need to be embedded respecting the toric action on M . The goal of this not...

We use toric geometry to study open string mirror symmetry on compact Calabi–Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs. We define a linear sigma model for the brane geo...

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of ...