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

研究代数簇的稳定性与极值卡勒度量的存在性之间的关系。

• 批准号: EP/D065933/1
• 负责人: Gabor Szekelyhidi
• 金额: $ 28.11万
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Fellowship
• 财政年份: 2006
• 资助国家: 英国
• 起止时间: 2006 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FD065933%2F1
• 关键词: Studying; relation; between; stability; algebraic

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.

差异几何形状中的一个旧问题是在歧管上找到杰出对象。对于Physicallinks，一种吸引人的方法正在试图找到最小化曲率功能的功能功能的指标。在复杂的歧管的情况下，这种方法是对极端指标的定义。这些是标量曲率的thel^2规范的关键点，该标量曲率的定义是根据GivencOhomology类中的指标空间定义的。或者，极端度量的标态曲率是圆形矢量场。至关重要的问题是决定何时存在极端指标。通过解释标量曲率ASA力矩图，我们已经能够做出精确的猜想来回答这个问题，但是证据仍然遥不可及。主要的猜想是，在且仅当它是k-stable时，代数品种承认一个极端指标。拟议的研究的一般目的是研究这个问题并证明了猜想的特殊情况。一种特殊的情况是感谢您的品种。这些具有最大尺寸圆环作用的面积品种，并且通过几何形状可以根据欧几里得空间中的组合和凸几何形状来研究它们。希望最终可以将其在此图案中取得的进展来在一般情况下取得进步。在曲曲面的情况下，涉及稳定标量曲率指标的较弱的猜想的证据已达到范围，在提议的研究中，我们旨在将结果推广到极端指标。对于一般品种不起作用的复的品种问题的一个方面是对凸functions的变化变性的简单描述。拟议的研究包括研究一般情况下的退化的计划，这是朝着福利案例扩展结果的一步。