Projective Varieties with Torus Action

Projective toric varieties are described either by lattice polytopes in the character group of the torus, or by a polyhedral fan. In the latter case, the projective embedding is encoded by a piecewise linear function on the fan. We will generalize this concept to the case of torus actions of smaller dimension such as the action of (C) on Grass(2, n). The resulting description includes a direct information about the position of the orbits inside the projective variety. We report results obtained together with Jürgen Hausen, Georg Hein, and Hendrik Süß, cf. [AlHa], [AHS], [AlHe]. According to them, normal varieties X with torus action can be described by divisors D on their Chow quotients Y . However, this requires the use of rather strange coefficients forD – they are polyhedra or polyhedral subdivisions. All coefficients have the same asymptotic behavior – visible as the socalled tail cone or fan of D. This language comprises that of toric varieties as well as the theory of (good) C actions. To describe some details, let T be an affine torus over an algebraically closed field K of characteristic 0. It gives rise to the mutually dual free abelian groups M := HomalgGrp(T,K ) and N := HomalgGrp(K , T ), and, via T = SpecK[M ], the torus can be recovered from them. Denote by NQ := N ⊗Z Q the corresponding vector space over Q. Definition 1. If σ ⊆ NQ is a polyhedral cone, then denote by Pol(NQ, σ) the Grothendieck group of the semigroup Pol(NQ, σ) := {∆ ⊆ NQ | ∆ = σ + [compact polytope]} with respect to Minkowski addition. Moreover, tail(∆) := σ is called the tail cone of the elements of Pol(NQ, σ). Let Y be a normal and semiprojective (i.e. Y → Y0 is projective over an affine Y0) K-variety. A Q-Cartier divisors on Y is called semiample if a multiple of it becomes base point free. Definition 2. An element D = ∑ i ∆i⊗Di ∈ Pol(NQ, σ)⊗ZCaDiv(Y ) with prime divisors Di is called a polyhedral divisor on (Y,N) with tail cone σ if ∆i ∈ Pol (NQ, σ) and if the evaluations D(u) := ∑ i min〈∆i, u〉Di are semiample for u ∈ σ ∨ ∩M and big for u ∈ int σ ∩M . (Note that the membership u ∈ σ := {u ∈MQ | 〈σ, u〉 ≥ 0} guarantees that min〈∆i, u〉 > −∞.) The common tail cone σ of the coefficients ∆i will be denoted by tail(D). The positivity assumptions imply that D(u) + D(u) ≤ D(u + u), hence OY (D) := ⊕u∈σ∨∩MOY (D(u)) becomes a sheaf of rings. So, our polyhedral divisor D gives rise