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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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References

1.Borel, A. and Lie, J., Compactifications of symmetric and locally symmetric spaces (Birkhauser 2006).Google Scholar
2.Boucksom, S., Hisamoto, T. and Jonsson, M., Uniform K-stability, Duistermaat–Heckman measures and singularities of pairs, Ann. Inst. Fourier 67(2) (2017), 743–841.Google Scholar
3.Boucksom, S., Hisamoto, T. and Jonsson, M., Uniform K-stability and asymptotics of energy functionals in Kähler geometry, JEMS, to appear.Google Scholar
4.Boucksom, S. and Jonsson, M., Singular semipositive metrics on line bundles on varieties over trivially valued fields, preprint (arXiv:1801.08229, 2018).Google Scholar
5.Bridson, M. and Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften, Volume 319 (Springer-Verlag, Berlin, 1999).Google Scholar
6.Codogni, G. and Dervan, R., Non-reductive automorphism groups, the Loewy filtration and K-stability, Ann. Inst. Fourier 66(5) (2016), 1895–1921.Google Scholar
7.Dervan, R., Uniform stability of twisted constant scalar curvature Kähler metrics, Int. Math. Res. Not. 15 2016), 4728–4783.Google Scholar
8.Dervan, R., Relative K-stability for Kähler manifolds, Math. Ann., to appear.Google Scholar
9.Dervan, R. and Ross, J., K-stability for Kähler manifolds, Math. Res. Lett. 24(3) (2017), 689–739.Google Scholar
10.Di Nezza, E. and Guedj, V., Geometry and topology of the space of Kähler metrics on singular varieties, Compos. Math. 154(8) (2018), 1593–1632.Google Scholar
11.Donaldson, S., Symmetric spaces, Kähler geometry and Hamiltonian dynamics, American Mathematical Society Translations, Series 2, Volume 196 (American Mathematical Society, Providence RI, 1999) pp. 13–33.Google Scholar
12.Donaldson, S. K., Scalar curvature and stability of toric varieties, J. Differential Geom. 62(2) (2002), 289–349.Google Scholar
13.Donaldson, S. K., Lower bounds of Calabi functionals, J. Differential Geom. 70(3) (2005), 453–472.Google Scholar
14.Kapovich, M., Leeb, B. and Millson, J., Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity, J. Differential Geom. 81 (2009), 297–354.Google Scholar
15.Kempf, G., Instability in invariant theory, Ann. of Math. (2) 108(2) (1978), 299–316.Google Scholar
16.Kleiner, B. and Leeb, B., Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, C. R. Acad. Sci. Paris Ser. 324(6) (1997), 639–643.Google Scholar
17.Lazarsfeld, R., Positivity in algebraic geometry, Volume 1 (Springer 2014).Google Scholar
18.Martin, F. and Gubler, W., On Zhang's semipositive metrics, preprint (arXiv:1608.08030, 2016).Google Scholar
19.Mumford, D., Fogarty, J. and Kirwan, F., Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 3rd edn, Volume 34 (Springer-Verlag, Berlin, 1994).Google Scholar
20.Odaka, Y., On parametrization, optimization and triviality of test configurations, Proc. Amer. Math. Soc. 143 (2015), 25–33.Google Scholar
21.Ross, J. and Thomas, R. P., A study of the Hilbert–Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), 201–255.Google Scholar
22.Rousseau, G., Immeubles spheriques et theorie des invariants C. R. Acad. Sci. Paris Ser. A-B 286 (1978), A247–A250.Google Scholar
23.Serre, J.-P., Complète réductibilité, Séminaire Bourbaki, Volume 2003/2004, Exposés 924–937, Astérisque (SMF, 2005) 195–217.Google Scholar
24.Sjöström Dyrefelt, Z., K-semistability of cscK manifolds with transcendental cohomology class, J. Geom. Anal. 28(4) (2018), 2927–2960.Google Scholar
25.Székelyhidi, G., Filtrations and test-configurations, Math. Ann. 362 (2015) 451.Google Scholar
26.Székelyhidi, G., Extremal Kähler metrics, Proceedings of the International Congress of Mathematicians, Volume II (Kyung Moon Sa Co. Ltd, Seoul, 2014).Google Scholar
27.Witt-Nystrom, D., Test configurations and Okounkov bodies, Compos. Math. 148(6) (2012), 1736–1756.Google Scholar