Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

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

Request Permissions   Purchase Content 


On Parametrization, optimization and triviality of test configurations

Author: Yuji Odaka
Journal: Proc. Amer. Math. Soc. 143 (2015), 25-33
MSC (2010): Primary 14L24; Secondary 32Q20
Published electronically: August 19, 2014
MathSciNet review: 3272728
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AB08] Peter Abramenko and Kenneth S. Brown, Buildings, Theory and applications, Graduate Texts in Mathematics, vol. 248, Springer, New York, 2008. MR 2439729 (2009g:20055)
  • [Don02] S. Donaldson, Scalar curvature and stability of toric varieties, J.Diff. Geom. 62 (2002), no. 2, 289-349. MR 1988506 (2005c:32028)
  • [Don05] S. K. Donaldson, Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), no. 3, 453-472. MR 2192937 (2006k:32045)
  • [Don10] S. Donaldson, Stability, birational transformations and the Kähler-Einstein problem, Surv. Differ. Geom., 17, Int. Press, Boston, MA, 2012. MR 3076062
  • [GSZ11] T. L. Gomez, I. Sols, and A. Zamora, A GIT interpretration of the Harder-Narasimhan filtration, arXiv:1112.1886.
  • [Kem78] George R. Kempf, Instability in invariant theory, Ann. of Math. (2) 108 (1978), no. 2, 299-316. MR 506989 (80c:20057),
  • [LX11] C. Li and C. Xu, Special test configurations and K-stability of $ \mathbb{Q}$-Fano varieties, arXiv:1111.5398.
  • [Mum65] David Mumford, Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Neue Folge, Band 34, Springer-Verlag, Berlin, 1965. MR 0214602 (35 #5451)
  • [Od11] Y. Odaka, A generalization of the Ross-Thomas slope theory, Osaka J. Math. 50 (2013), no. 1, 171-185. MR 3080636
  • [Rou78] 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 0506257 (58 #22063)
  • [RT07] 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 (2007k:14091),
  • [Stp11] J. Stoppa, A note on the definition of K-stability, arXiv:1111.5826.
  • [Sze08] Gábor Székelyhidi, Optimal test-configurations for toric varieties, J. Differential Geom. 80 (2008), no. 3, 501-523. MR 2472481 (2009m:53179)
  • [Sze11] G. Szekelyhidi, Filtrations and Test-configurations, arXiv:1111.4986.
  • [Tia97] Gang Tian, Kähler-Einstein metrics with positive scalar curvature, Invent. Math. 130 (1997), no. 1, 1-37. MR 1471884 (99e:53065),
  • [Tit74] Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin, 1974. MR 0470099 (57 #9866)
  • [Wan08] X. Wang, Height and GIT weight, Math. Res. Lett. 19 (2012), no. 4, 909-926. MR 3008424

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

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
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society