ATAG_Algebraic Torus Actions: Geometry and Combinatorics

具有代数环面作用的代数簇理论是代数几何、拓扑学、表示论和离散数学等学科边缘的一个广阔而活跃的研究领域。本提案旨在扩展与等变上同调、环面几何和准齐次空间理论相关建立的方法的适用性，以理解具有环面作用的代数簇的几何。我们主要感兴趣的情况下，环面行动有一个有限数量的不动点和一维轨道。网格是Goresky-Kottwitz-MacPherson图的改进版本，而p-因子来自环面几何，将用于积累关于这些变体的数据，这些变体是关于作用环面的特征空间（或其对偶空间）中的对象的。预计凸和热带几何以及组合方法将提供处理和分析这些数据的工具。作为一个结果，我们期望在极小模型程序和分类的Fano流形的新的结果。研究的对象将是GIT和Chow定理，Hilbert格式和具有环面作用的簇上的线性系统。此外，奇异性来自Fano-Mori收缩，封闭的作用轨道和退化的光滑品种与环面作用将被研究。

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.