Osaka Journal of Mathematics

Chow-stability and Hilbert-stability in Mumford's geometric invariant theory

Toshiki Mabuchi

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Abstract

In this note, we shall show that Chow-stability and Hilbert-stability in GIT asymptotically coincide. The proof in [5] is simplified in the present form, while a quick review is in [6].

Article information

Source
Osaka J. Math., Volume 45, Number 3 (2008), 833-846.

Dates
First available in Project Euclid: 17 September 2008

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1221656656

Mathematical Reviews number (MathSciNet)
MR2468597

Zentralblatt MATH identifier
1156.14039

Subjects
Primary: 14L24: Geometric invariant theory [See also 13A50] 32Q15: Kähler manifolds
Secondary: 32Q20: Kähler-Einstein manifolds [See also 53Cxx] 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)

Citation

Mabuchi, Toshiki. Chow-stability and Hilbert-stability in Mumford's geometric invariant theory. Osaka J. Math. 45 (2008), no. 3, 833--846. https://projecteuclid.org/euclid.ojm/1221656656


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References

  • S.K. Donaldson: Scalar curvature and stability of toric varieties, J. Differential Geom. 62 (2002), 289--349.
  • S.K. Donaldson: Scalar curvature and projective embeddings, II, Q.J. Math. 56 (2005), 345--356.
  • S.K. Donaldson: Lower bounds on the Calabi functional, J. Differential Geom. 70 (2005), 453--472.
  • J. Fogarty: Truncated Hilbert functors, J. Reine Angew. Math. 234 (1969), 65--88.
  • T. Mabuchi: The Chow-stability and Hilbert-stability in Mumford's geometric invariant theory, arXiv:math.DG/0607590 v1 (2006); v2 (2007).
  • T. Mabuchi: Chow stability and Hilbert stability from differential geometric viewpoints; in Complex Geometry in Osaka, Lect. Note Ser. in Math. 6, Osaka Math. Publ., 2008, 1--9.
  • D. Mumford: Varieties defined by quadratic equations; in Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969), Ed. Cremonese, Rome, 1970, 29--100.
  • D. Mumford: Stability of projective varieties, Enseignement Math. (2) 23 (1977), 39--110.
  • D. Mumford, J. Fogarty and F. Kirwan: Geometric Invariant Theory, third edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) 34, Springer, Berlin, 1994.
  • Y. Matsushima: Holomorphic Vector Fields on Compact Kähler Manifolds, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics 7, Amer. Math. Soc., Providence, R.I., 1971.
  • Y. Odaka: The GIT-stability of polarised varieties via discrepancy, arXiv:math.AG/ 0807.1716 (2008).
  • S.T. Paul: Geometric analysis of Chow Mumford stability, Adv. Math. 182 (2004), 333--356.
  • S.T. Paul and G. Tian: Algebraic and analytic K-stability, arXiv:math.DG/0405530 (2004).