An Introduction to T-Varieties

Toric varieties generalize naturally to so-called complexity k T-varieties, that is, normal varieties with an effective codimension-k torus action.  Altmann, Hausen, and Suess have given a description of these varieties  in terms of a k-dimensional variety along with combinatorial data called a divisorial fan.  More recent results concerning T-varieties focus on the complexity-one case and include descriptions of divisors, intersection numbers, deformations, and applications in coding theory. In my talk, I will focus on the basic construction of T-varieties from divisorial fans and survey some of the more recent results.