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寻找流形上的可分辨度量。对于这个问题，一个很吸引人的方法，对于物理链路也是如此，就是试图找到一个最小化函数的度量，通常是曲率的函数。在复杂流形的情况下，这种方法导致极值度量的定义。这些是标量曲率的theL^2范数的临界点，定义在给定上同调类的度量空间上。另一个极值度规是这样的，它的标量曲率的梯度是一个全纯向量场。关键问题是确定何时存在极值度量。通过将标量曲率解释为矩图，我们已经能够做出精确的推测来回答这个问题，但是证明仍然遥不可及。主要的猜想是一个代数变量当且仅当它是k稳定的时才允许一个极值度量。这个条件是纯代数几何性质的，而极值度量的存在性是微分几何和微分分析中的一个问题。本研究的总体目的是研究这一问题，并证明猜想的特殊情况。一个特别的例子是托利豆的品种。这些是具有极大维环面作用的代数变体，通过几何约简，可以用欧几里德空间中多面体的组合学和凸几何来研究它们。希望在这种简单情况下取得的进展最终可以用于在一般情况下取得进展。在环面的情况下，一个弱猜想的证明，它处理常数标量曲率度量，是触手可及的，在提议的研究中，我们的目标是将结果推广到极值度量。环型变异体不适用于一般变异体的问题的一个方面是用凸函数来简单描述变异体的退化。拟议的研究包括研究一般情况下的退化，作为扩展环形情况结果的一步。

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.