Skip to Main Content

Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

On Parametrization, optimization and triviality of test configurations
HTML articles powered by AMS MathViewer

by Yuji Odaka PDF
Proc. Amer. Math. Soc. 143 (2015), 25-33 Request permission

Abstract:

We give a parametrization of test configurations in the sense of Donaldson via spherical buildings, and show the existence of “optimal” destabilizing test configurations for unstable varieties, in the wake of Mumford and Kempf. We also give an account of the recent slight amendment to the definition of K-stability after Li-Xu, from two other viewpoints: from the one-parameter subgroups and from the author’s blow-up formalism.
References
  • Peter Abramenko and Kenneth S. Brown, Buildings, Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008. Theory and applications. MR 2439729, DOI 10.1007/978-0-387-78835-7
  • S. K. Donaldson, Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), no. 2, 289–349. MR 1988506
  • S. K. Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), no. 3, 453–472. MR 2192937
  • S. K. Donaldson, Stability, birational transformations and the Kahler-Einstein problem, Surveys in differential geometry. Vol. XVII, Surv. Differ. Geom., vol. 17, Int. Press, Boston, MA, 2012, pp. 203–228. MR 3076062, DOI 10.4310/SDG.2012.v17.n1.a5
  • T. L. Gomez, I. Sols, and A. Zamora, A GIT interpretration of the Harder-Narasimhan filtration, arXiv:1112.1886.
  • George R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299–316. MR 506989, DOI 10.2307/1971168
  • C. Li and C. Xu, Special test configurations and K-stability of $\mathbb {Q}$-Fano varieties, arXiv:1111.5398.
  • David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 34, Springer-Verlag, Berlin-New York, 1965. MR 0214602
  • Yuji Odaka, A generalization of the Ross-Thomas slope theory, Osaka J. Math. 50 (2013), no. 1, 171–185. MR 3080636
  • Guy Rousseau, Immeubles sphériques et théorie des invariants, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 5, A247–A250 (French, with English summary). MR 506257
  • Julius Ross and Richard Thomas, A study of the Hilbert-Mumford criterion for the stability of projective varieties, J. Algebraic Geom. 16 (2007), no. 2, 201–255. MR 2274514, DOI 10.1090/S1056-3911-06-00461-9
  • J. Stoppa, A note on the definition of K-stability, arXiv:1111.5826.
  • Gábor Székelyhidi, Optimal test-configurations for toric varieties, J. Differential Geom. 80 (2008), no. 3, 501–523. MR 2472481
  • G. Szekelyhidi, Filtrations and Test-configurations, arXiv:1111.4986.
  • Gang Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1–37. MR 1471884, DOI 10.1007/s002220050176
  • Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
  • Xiaowei Wang, Height and GIT weight, Math. Res. Lett. 19 (2012), no. 4, 909–926. MR 3008424, DOI 10.4310/MRL.2012.v19.n4.a14
Similar Articles
  • Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 14L24, 32Q20
  • Retrieve articles in all journals with MSC (2010): 14L24, 32Q20
Additional Information
  • Yuji Odaka
  • Affiliation: Department of Mathematics, Kyoto University, Kyoto, 606-850, Japan
  • Email: yodaka@math.kyoto-u.ac.jp
  • Received by editor(s): March 17, 2012
  • Received by editor(s) in revised form: February 26, 2013
  • Published electronically: August 19, 2014
  • Additional Notes: The author was partially supported by the Grant-in-Aid for Scientific Research (KAKENHI No. 21-3748) and the Grant-in-Aid for JSPS fellows (PD)
  • Communicated by: Lev Borisov
  • © Copyright 2014 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 143 (2015), 25-33
  • MSC (2010): Primary 14L24; Secondary 32Q20
  • DOI: https://doi.org/10.1090/S0002-9939-2014-12204-0
  • MathSciNet review: 3272728