Toric sheaves and polyhedra

TL;DR

This paper explores the relationship between toric sheaves on smooth projective toric varieties and polyhedral geometry, providing explicit constructions and cohomology computations using polyhedral methods.

Contribution

It introduces a polyhedral approach to studying toric sheaves, including explicit constructions of universal extensions and a spectral sequence for cohomology calculation.

Findings

Constructed the torus invariant universal extension of nef line bundles.

Linked cohomology of toric sheaves to reduced cohomology of polyhedral subsets.

Developed a double complex and spectral sequence for cohomology computation.

Abstract

Over a smooth projective toric variety we study toric sheaves, that is, reflexive sheaves equivariant with respect to the acting torus, from a polyhedral point of view. One application is the explicit construction of the torus invariant universal extension of two nef line bundles via polyhedral inclusion/exclusion sequences. Second, we link the cohomology of toric sheaves to the cohomology of certain constructible sheaves explicitly built out of the associated polyhedra. For the latter we define a concrete double complex and a spectral sequence which computes the cohomology of toric sheaves from the reduced cohomology of polyhedral subsets living in the realification of the character lattice of the toric variety.