ATAG_Algebraic Torus Actions: Geometry and Combinatorics

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The theory of algebraic varieties with algebraic torus action is a vast and active research field on the border of algebraic geometry,topology, representation theory and discrete mathematics. The present proposal aims at extending the applicability of methodsestablished in relation to equivariant cohomology, toric geometry and theory of quasi-homogeneous spaces in order to understand the geometry of algebraic varieties with torus action. We are primarily interested in the situation where the torus action has a finite number of fixed points and of one dimensional orbits. Grids, which are refined versions of Goresky-Kottwitz-MacPherson graphs,and p-divisors, stemming from toric geometry, will be used to accumulate the data about these varieties in terms of objects living in the space of characters (or its dual) of the acting torus. Convex and tropical geometry as well as combinatorial methods are expected to provide the tools to proceed and analyze this data. As an outcome we expect new results in minimal model program and classification of Fano manifolds. The objects under study will be GIT and Chow quotients, Hilbert schemes and linear systems on varieties with torus action. Moreover, the singularities coming from Fano-Mori contractions, closures of orbits of the action and degenerations of smooth varieties with torus action will be studied.