An example of an asymptotically Chow unstable manifold with constant scalar curvature

Ono, Hajime; Sano, Yuji; Yotsutani, Naoto

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 $c_1\left(L\right)$, et si son groupe d’automorphismes $\mathrm{Aut}(V,L)$ 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 $\mathrm{Aut}(V,L)$ n’est pas discret.

Donaldson proved that if a polarized manifold $(V,L)$ has constant scalar curvature Kähler metrics in $c_1\left(L\right)$ and its automorphism group $\mathrm{Aut}(V,L)$ 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 $\mathrm{Aut}(V,L)$ is not discrete.

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

[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