ATAG_Algebraic Torus Actions: Geometry and Combinatorics

具有代数环面作用的代数变分理论是一个处于代数几何、拓扑学、表示理论和离散数学边缘的广阔而活跃的研究领域。本文旨在推广在等变上同调、环面几何和拟齐次空间理论中建立的方法的适用性，以理解具有环面作用的代数变异的几何。我们主要对环面作用有有限个数的固定点和一维轨道的情况感兴趣。网格是Goresky-Kottwitz-MacPherson图的改进版本，p-除数源于环面几何，将被用来积累这些变化的数据，这些数据是根据生活在作用环面的字符（或其对偶）空间中的对象来计算的。凸几何和热带几何以及组合方法有望提供工具来处理和分析这些数据。期望在最小模型规划和法诺流形分类方面取得新的成果。研究对象包括GIT商和Chow商、Hilbert格式和环面作用下的线性系统。此外，还研究了由Fano-Mori收缩引起的奇异性、作用的轨道闭包以及环面作用下光滑变异的退化。

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.