An example of an asymptotically Chow unstable manifold with constant scalar curvature
[Un exemple de variété à courbure scalaire constante asymptotiquement instable au sens de Chow]
Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1265-1287.

Donaldson a prouvé que, si une variété polarisée (V,L) admet une métrique kählérienne à courbure scalaire constante dans c 1 (L), et si son groupe d’automorphismes Aut(V,L) est discret, alors (V,L) est asymptotiquement stable au sens de Chow. Dans cet article, nous allons montrer un exemple qui implique que le résultat ci-dessus ne s’étend pas au cas où Aut(V,L) n’est pas discret.

Donaldson proved that if a polarized manifold (V,L) has constant scalar curvature Kähler metrics in c 1 (L) and its automorphism group Aut(V,L) is discrete, (V,L) is asymptotically Chow stable. In this paper, we shall show an example which implies that the above result does not hold in the case where Aut(V,L) is not discrete.

DOI : 10.5802/aif.2722
Classification : 53C55, 53C21, 55N91
Keywords: asymptotic Chow stability, Kähler metric of constant scalar curvature, toric Fano manifold, Futaki invariant
Mot clés : stabilité asymptotique au sens de Chow, métrique kählérienne à courbure scalaire contsante, variété de Fano torique, invariant de Futaki
Ono, Hajime 1 ; Sano, Yuji 2 ; Yotsutani, Naoto 3

1 Tokyo University of Science Faculty of Science and Technology Department of Mathematics 2641 Yamazaki, Noda Chiba 278-8510 (Japan)
2 Kumamoto University Graduate School of Science and Technology 2-39-1, Kurokami Kumamoto, 860-8555 (Japan)
3 University of Science and Technology of China School of Mathematics Hefei, Anhui 230026 P.R. (China)
@article{AIF_2012__62_4_1265_0,
     author = {Ono, Hajime and Sano, Yuji and Yotsutani, Naoto},
     title = {An example of an asymptotically {Chow} unstable manifold  with constant scalar curvature},
     journal = {Annales de l'Institut Fourier},
     pages = {1265--1287},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {4},
     year = {2012},
     doi = {10.5802/aif.2722},
     zbl = {1255.53057},
     mrnumber = {3025743},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2722/}
}
TY  - JOUR
AU  - Ono, Hajime
AU  - Sano, Yuji
AU  - Yotsutani, Naoto
TI  - An example of an asymptotically Chow unstable manifold  with constant scalar curvature
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 1265
EP  - 1287
VL  - 62
IS  - 4
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2722/
DO  - 10.5802/aif.2722
LA  - en
ID  - AIF_2012__62_4_1265_0
ER  - 
%0 Journal Article
%A Ono, Hajime
%A Sano, Yuji
%A Yotsutani, Naoto
%T An example of an asymptotically Chow unstable manifold  with constant scalar curvature
%J Annales de l'Institut Fourier
%D 2012
%P 1265-1287
%V 62
%N 4
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2722/
%R 10.5802/aif.2722
%G en
%F AIF_2012__62_4_1265_0
Ono, Hajime; Sano, Yuji; Yotsutani, Naoto. An example of an asymptotically Chow unstable manifold  with constant scalar curvature. Annales de l'Institut Fourier, Tome 62 (2012) no. 4, pp. 1265-1287. doi : 10.5802/aif.2722. http://www.numdam.org/articles/10.5802/aif.2722/

[1] Arezzo, Claudio; Loi, Andrea Moment maps, scalar curvature and quantization of Kähler manifolds, Comm. Math. Phys., Volume 246 (2004) no. 3, pp. 543-559 | DOI | MR | Zbl

[2] Batyrev, Victor V.; Selivanova, Elena N. Einstein-Kähler metrics on symmetric toric Fano manifolds, J. Reine Angew. Math., Volume 512 (1999), pp. 225-236 | DOI | MR | Zbl

[3] Donaldson, S. K. Scalar curvature and projective embeddings. I, J. Differential Geom., Volume 59 (2001) no. 3, pp. 479-522 http://projecteuclid.org/getRecord?id=euclid.jdg/1090349449 | MR | Zbl

[4] Donaldson, S. K. Scalar curvature and stability of toric varieties, J. Differential Geom., Volume 62 (2002) no. 2, pp. 289-349 http://projecteuclid.org/getRecord?id=euclid.jdg/1090950195 | MR | Zbl

[5] Futaki, A. An obstruction to the existence of Einstein Kähler metrics, Invent. Math., Volume 73 (1983) no. 3, pp. 437-443 | DOI | MR | Zbl

[6] Futaki, A. Kähler-Einstein metrics and integral invariants, Lecture Notes in Mathematics, 1314, Springer-Verlag, Berlin, 1988 | MR | Zbl

[7] Futaki, A. Asymptotic Chow semi-stability and integral invariants, Internat. J. Math., Volume 15 (2004) no. 9, pp. 967-979 | DOI | MR | Zbl

[8] Futaki, A.; Morita, S. Invariant polynomials of the automorphism group of a compact complex manifold, J. Differential Geom., Volume 21 (1985) no. 1, pp. 135-142 http://projecteuclid.org/getRecord?id=euclid.jdg/1214439469 | MR | Zbl

[9] Futaki, A.; Ono, H.; Sano, Y. Hilbert series and obstructions to asymptotic semistability, Adv. Math., Volume 226 (2011) no. 1, pp. 254-284 | DOI | MR | Zbl

[10] Mabuchi, Toshiki An obstruction to asymptotic semistability and approximate critical metrics, Osaka J. Math., Volume 41 (2004) no. 2, pp. 463-472 http://projecteuclid.org/getRecord?id=euclid.ojm/1153493522 | MR | Zbl

[11] Mabuchi, Toshiki An energy-theoretic approach to the Hitchin-Kobayashi correspondence for manifolds. I, Invent. Math., Volume 159 (2005) no. 2, pp. 225-243 | DOI | MR | Zbl

[12] Miller, Ezra; Sturmfels, Bernd Combinatorial commutative algebra, Graduate Texts in Mathematics, 227, Springer-Verlag, New York, 2005 | MR | Zbl

[13] Mumford, D.; Fogarty, J.; Kirwan, F. Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], 34, Springer-Verlag, Berlin, 1994 | MR | Zbl

[14] Nill, B.; Paffenholz, A. Examples of non-symmetric Kähler-Einstein toric Fano manifolds (http://arxiv.org/abs/0905.2054)

[15] Oda, Tadao Convex bodies and algebraic geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 15, Springer-Verlag, Berlin, 1988 (An introduction to the theory of toric varieties, Translated from the Japanese) | MR | Zbl

[16] Ono, H. A necessary condition for Chow semistability of polarized toric manifolds, J. Math. Soc., Japan, Volume 63 (2011) no. 4, pp. 1377-1389 | DOI | MR | Zbl

[17] Tian, Gang Kähler-Einstein metrics with positive scalar curvature, Invent. Math., Volume 130 (1997) no. 1, pp. 1-37 | DOI | MR | Zbl

[18] Wang, Xu-Jia; Zhu, Xiaohua Kähler-Ricci solitons on toric manifolds with positive first Chern class, Adv. Math., Volume 188 (2004) no. 1, pp. 87-103 | DOI | MR | Zbl

[19] Yau, Shing Tung Calabi’s conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A., Volume 74 (1977) no. 5, pp. 1798-1799 | DOI | MR | Zbl

[20] Yau, Shing-Tung Perspectives on geometric analysis, Surveys in differential geometry. Vol. X (Surv. Differ. Geom.), Volume 10, Int. Press, Somerville, MA, 2006, pp. 275-379 | MR | Zbl

Cité par Sources :