The Geometry of T-Varieties

TL;DR

This paper surveys the use of polyhedral divisors in describing T-varieties, drawing parallels with toric varieties and covering topics like singularities, divisors, cohomology, and deformations.

Contribution

It provides a comprehensive overview of the polyhedral divisor language for T-varieties, connecting it to classical toric geometry and expanding on various geometric aspects.

Findings

Unified framework for T-varieties via polyhedral divisors

Insights into singularities and properness of T-varieties

Discussion of cohomology and deformation theory in this context

Abstract

This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.