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Tits Buildings and K-Stability

Part of: Algebraic groups Complex manifolds

Published online by Cambridge University Press:  30 January 2019

Giulio Codogni
Giulio Codogni*
Affiliation:
EPFL, SB MATHGEOM CAG, MA B3 635 (Bâtiment MA), Station 8, CH-1015, Lausanne, Switzerland ([email protected])
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Abstract

A polarized variety is K-stable if, for any test configuration, the Donaldson–Futaki invariant is positive. In this paper, inspired by classical geometric invariant theory, we describe the space of test configurations as a limit of a direct system of Tits buildings. We show that the Donaldson–Futaki invariant, conveniently normalized, is a continuous function on this space. We also introduce a pseudo-metric on the space of test configurations. Recall that K-stability can be enhanced by requiring that the Donaldson–Futaki invariant is positive on any admissible filtration of the co-ordinate ring. We show that admissible filtrations give rise to Cauchy sequences of test configurations with respect to the above mentioned pseudo-metric.

Keywords

K-stabilityTits buildingsgeometric invariant theoryYau–Tian–Donaldson conjectureconstant scalar curvature Kähler metricsKähler–Einstein metrics

MSC classification

Primary: 14L24: Geometric invariant theory
Secondary: 32Q20: Kähler-Einstein manifolds
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 3 , August 2019 , pp. 799 - 815
Copyright
Copyright © Edinburgh Mathematical Society 2019 

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