Studying the relation between stability of algebraic varieties and the existence of extremal Kahler metrics.

微分几何中的一个老问题是求流形上的非线性度量。一个有吸引力的方法来解决这个问题，物理链接，以及，是试图找到度量，最小化的功能，normallya功能的曲率。在复流形的情况下，这种方法导致极值度量的定义。这些是定义在给定上同调类中的度量空间上的标量曲率的L^2范数的临界点。另一种极端度量是这样的，它的标量曲率的梯度是一个全纯向量场。关键的问题是决定何时存在极值度量。通过将标量曲率解释阿萨矩映射，我们已经能够对这个问题做出精确的解答，但证明仍然遥不可及。主要的猜想是代数簇存在极值度量当且仅当它是K-稳定的，这个条件是纯代数几何的，而极值度量的存在性是微分几何和分析中的一个问题。一般的目的，拟议的研究是研究这个问题，并证明特殊情况下的结构。一个特殊的例子是复曲面品种。这些代数簇具有极大维环面作用，通过几何约化，人们可以用欧氏空间中多面体的组合学和凸几何学来研究它们。我们希望在这个简单的例子中所取得的进展最终可以被用于在一般情况下取得进展。在复曲面的情况下，证明一个较弱的猜想，其中涉及常数标量曲率度量，是触手可及的，在拟议的研究，我们的目标是推广到极值度量的结果。一个方面的问题环面品种不适用于一般品种是简单的描述退化的各种凸函数。拟议的研究包括计划研究退化的一般情况下，作为一个步骤，扩大结果从复曲面的情况。

An old problem in differential geometry is that of finding distinguishedmetrics on manifolds. An appealing approach to this problem with, physicallinks as well, is trying to find metrics which minimise a functional, normallya function of the curvature. In the case of complex manifolds, this approachleads to the definition of extremal metrics. These are critical points of theL^2 norm of the scalar curvature, defined on the space of metrics in a givencohomology class. Alternatively an extremal metric is such that the gradientof its scalar curvature is a holomorphic vector field. The crucial question isto decide when extremal metrics exist. By interpreting the scalar curvature asa moment map, we have been able to make precise conjectures to answer thisquestion, but the proofs are still out of reach. The main conjecture is thatan algebraic variety admits an extremal metrics if and only if it is K-stable.This condition is purely algebro-geometric whereas the existence of extremalmetrics is a question in differential geometry and analysis. The general aimof the proposed research is to study this problem and prove special cases ofthe conjectures. One particular case is that of toric varieties. These arealgebraic varieties with a maximal dimensional torus action, and by geometricreduction one can study them in terms of the combinatorics and convex geometryof polytopes in Euclidean space. The hope is that progress made in thissimpler case can eventually be used to make advances in the general case. Inthe case of toric surfaces a proof of a weaker conjecture, which deals withconstant scalar curvature metrics, is within reach and in the proposedresearch we aim to generalise that result to extremal metrics. One aspect ofthe problem for toric varieties which does not work for general varieties isthe simple description of degenerations of the variety in terms of convexfunctions. The proposed research includes plans to study degenerations in thegeneral case as a step towards extending results from the toric case.