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.

微分几何中的一个老问题是在流形上找到可区分的度量。解决这个问题的一个有吸引力的方法（物理链接也是如此）是试图找到最小化函数（通常是曲率函数）的度量。在复杂流形的情况下，这种方法导致了极值度量的定义。这些是标量曲率的 L^2 范数的临界点，在给定上同调类的度量空间上定义。或者，极值度量使得其标量曲率的梯度是全纯矢量场。关键问题是确定何时存在极值指标。通过将标量曲率解释为矩图，我们已经能够做出精确的猜想来回答这个问题，但证明仍然遥不可及。主要猜想是，代数簇当且仅当它是 K 稳定时才承认极值度量。这个条件是纯粹的代数几何，而极值度量的存在是微分几何和分析中的一个问题。本研究的总体目的是研究这个问题并证明猜想的特殊情况。一种特殊情况是复曲面品种。这些是具有最大维环面作用的代数簇，通过几何约简，人们可以根据欧几里得空间中多胞体的组合学和凸几何来研究它们。希望在这个更简单的案例中取得的进展最终能够用于在一般案例中取得进展。在复曲面的情况下，处理恒定标量曲率度量的较弱猜想的证明是可以实现的，在拟议的研究中，我们的目标是将该结果推广到极值度量。环面簇问题的一个方面（不适用于一般簇）是用凸函数来简单描述簇的退化。拟议的研究包括研究一般情况下的退化的计划，作为扩展复曲面情况结果的一步。