The well-known connection between toric varieties and convex polyhedral objects has led to a fairly complete understanding of the geometry of toric varieties in combinatorial terms. A little over a decade ago, Altmann and Hausen developed a theory generalizing this connection to so-called T-varieties: normal varieties X endowed with an effective action of an algebraic torus T. Examples of such varieties include spherical varieties and toric vector bundles, among other examples. The situation where the dimension of X equals that of T is exactly the toric case.

The price to be paid for this generalization is that the theory of T-varieties is no longer purely combinatorial. Instead, a T-variety X may be described in terms of a quotient variety Y, endowed with combinatorial information in the form of a “polyhedral divisor” (in the affine case) or a “divisorial fan” (in general). The dimension of the quotient variety is the codimension of T in X; in the toric case, the quotient collapses to a point, and only the combinatorial information remains. In situations where the geometry of the quotient variety Y is relatively accessible (e.g. Y is a curve or projective space), many results concerning toric varieties can be generalized.

In this series of lectures, I will give an introduction to the Altmann-Hausen theory of polyhedral divisors. After outlining the general theory, I will focus on the case of complexity-one actions with two applications in mind: proving that rank two toric vector bundles are Mori Dream Spaces, and studying the deformation theory of toric varieties.