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

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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.