Tits Buildings and K-Stability. Issue 3 (30th January 2019)

• Record Type: Journal Article
• Authors: Codogni, Giulio
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 62:Issue 3(2019)
• Page Start: 799
• Page End: 815
• Publication Date: 2019-01-30
• Subjects: K-stability, -- Tits buildings, -- geometric invariant theory, -- Yau–Tian–Donaldson conjecture, -- constant scalar curvature Kähler metrics, -- Kähler–Einstein metrics; Primary 14L24, -- Secondary 32Q20; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091518000512
• Languages: English
• ISSNs: 0013-0915

Abstract

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.