Convergence of Bergman geodesics on CP 1
[Convergence des géodésiques de Bergman sur CP 1 ]
Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2209-2237.

L’espace des métriques de Kähler dans une classe donnée sur une variété projective kählérienne X est un espace symétrique de dimension infinie dont les géodésiques ω t sont des solutions d’une équation Monge-Ampère complexe homogène sur A×X, ou A={z:e -1 <|z|<1} . Phong-Sturm ont prouvé que les géodésiques Monge-Ampère des potentiels kählériens ϕ(t,z) de ω t peuvent être approximées dans un sens faible C 0 par géodésiques ϕ N (t,z) de l’espace symétrique de métriques de Bergman de hauteur N. Le but de cet article est de prouver que ϕ N (t,z)ϕ(t,z) dans C 2 ([0,1]×X) dans le cas des métriques toriques sur X=CP 1 .

The space of Kähler metrics in a fixed Kähler class on a projective Kähler manifold X is an infinite dimensional symmetric space whose geodesics ω t are solutions of a homogeneous complex Monge-Ampère equation in A×X, where A is an annulus. Phong-Sturm have proven that the Monge-Ampère geodesic of Kähler potentials ϕ(t,z) of ω t may be approximated in a weak C 0 sense by geodesics ϕ N (t,z) of the finite dimensional symmetric space of Bergman metrics of height N. In this article we prove that ϕ N (t,z)ϕ(t,z) in C 2 ([0,1]×X) in the case of toric Kähler metrics on X=CP 1 .

DOI : 10.5802/aif.2332
Classification : 53C55
Keywords: Bergman metric, Monge-Ampère equation, Bergman-Szegö kernel, toric metric, Kähler potential, symplectic potential
Mot clés : métrique de Bergman, équation Monge-Ampère, noyau de Bergman-Szegö, métrique torique, potential kählérien, potential symplectique
Song, Jian 1 ; Zelditch, Steve 2

1 Rutgers, The State University of New Jersey Department of Mathematics Hill Center-Busch Campus 110 Frelinghuysen Rd Piscataway, NJ 08854-8019 (USA)
2 Johns Hopkins University Department of Mathematics Baltimore, MD 21218 (USA)
@article{AIF_2007__57_7_2209_0,
     author = {Song, Jian and Zelditch, Steve},
     title = {Convergence of {Bergman} geodesics on $\mathbf{CP}^1$},
     journal = {Annales de l'Institut Fourier},
     pages = {2209--2237},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {57},
     number = {7},
     year = {2007},
     doi = {10.5802/aif.2332},
     zbl = {1144.53089},
     mrnumber = {2394541},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2332/}
}
TY  - JOUR
AU  - Song, Jian
AU  - Zelditch, Steve
TI  - Convergence of Bergman geodesics on $\mathbf{CP}^1$
JO  - Annales de l'Institut Fourier
PY  - 2007
SP  - 2209
EP  - 2237
VL  - 57
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2332/
DO  - 10.5802/aif.2332
LA  - en
ID  - AIF_2007__57_7_2209_0
ER  - 
%0 Journal Article
%A Song, Jian
%A Zelditch, Steve
%T Convergence of Bergman geodesics on $\mathbf{CP}^1$
%J Annales de l'Institut Fourier
%D 2007
%P 2209-2237
%V 57
%N 7
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2332/
%R 10.5802/aif.2332
%G en
%F AIF_2007__57_7_2209_0
Song, Jian; Zelditch, Steve. Convergence of Bergman geodesics on $\mathbf{CP}^1$. Annales de l'Institut Fourier, Tome 57 (2007) no. 7, pp. 2209-2237. doi : 10.5802/aif.2332. http://www.numdam.org/articles/10.5802/aif.2332/

[1] Abreu, Miguel Kähler geometry of toric manifolds in symplectic coordinates, Symplectic and contact topology: interactions and perspectives (Toronto, ON/Montreal, QC, 2001) (Fields Inst. Commun.), Volume 35, Amer. Math. Soc., Providence, RI, 2003, pp. 1-24 | MR | Zbl

[2] Arezzo, Claudio; Tian, Gang Infinite geodesic rays in the space of Kähler potentials, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), Volume 2 (2003) no. 4, pp. 617-630 | Numdam | MR

[3] Berndtsson, B. Positivity of direct image bundles and convexity on the space of Kähler metrics (preprint, arxiv: math.CV/0608385)

[4] Biskup, M.; Borgs, C.; Chayes, J. T.; Kleinwaks, L. J.; Kotecký, R. Partition function zeros at first-order phase transitions: a general analysis, Comm. Math. Phys., Volume 251 (2004) no. 1, pp. 79-131 | DOI | MR | Zbl

[5] Calabi, Eugenio Extremal Kähler metrics, Seminar on Differential Geometry (Ann. of Math. Stud.), Volume 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 259-290 | MR | Zbl

[6] Chen, X.; Tian, G. Geometry of Kähler metrics and foliations by discs (preprint, math.DG/0409433)

[7] Chen, Xiuxiong The space of Kähler metrics, J. Differential Geom., Volume 56 (2000) no. 2, pp. 189-234 | MR | Zbl

[8] Donaldson, S. K. Some numerical results in complex differential geometry (preprint, math.DG/0512625)

[9] Donaldson, S. K. Symmetric spaces, Kähler geometry and Hamiltonian dynamics, Northern California Symplectic Geometry Seminar (Amer. Math. Soc. Transl. Ser. 2), Volume 196, Amer. Math. Soc., Providence, RI, 1999, pp. 13-33 | MR | Zbl

[10] Donaldson, S. K. Scalar curvature and projective embeddings. I, J. Differential Geom., Volume 59 (2001) no. 3, pp. 479-522 | MR | Zbl

[11] Donaldson, S. K. Holomorphic discs and the complex Monge-Ampère equation, J. Symplectic Geom., Volume 1 (2002) no. 2, pp. 171-196 | MR | Zbl

[12] Donaldson, S. K. Scalar curvature and stability of toric varieties, J. Differential Geom., Volume 62 (2002) no. 2, pp. 289-349 | MR | Zbl

[13] Ellis, Richard S. Entropy, large deviations, and statistical mechanics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 271, Springer-Verlag, New York, 1985 | MR | Zbl

[14] Fulton, William Introduction to toric varieties, Annals of Mathematics Studies, 131, Princeton University Press, Princeton, NJ, 1993 (, The William H. Roever Lectures in Geometry) | MR | Zbl

[15] Guan, Daniel On modified Mabuchi functional and Mabuchi moduli space of Kähler metrics on toric bundles, Math. Res. Lett., Volume 6 (1999) no. 5-6, pp. 547-555 | MR | Zbl

[16] Guillemin, Victor Kähler structures on toric varieties, J. Differential Geom., Volume 40 (1994) no. 2, pp. 285-309 | MR | Zbl

[17] Hörmander, Lars The analysis of linear partial differential operators. I, Springer Study Edition, Springer-Verlag, Berlin, 1990 (Distribution theory and Fourier analysis) | MR | Zbl

[18] Mabuchi, Toshiki Some symplectic geometry on compact Kähler manifolds. I, Osaka J. Math., Volume 24 (1987) no. 2, pp. 227-252 | MR | Zbl

[19] Phong, D. H.; Sturm, Jacob The Monge-Ampère operator and geodesics in the space of Kähler potentials, Invent. Math., Volume 166 (2006) no. 1, pp. 125-149 | DOI | MR | Zbl

[20] Semmes, Stephen Complex Monge-Ampère and symplectic manifolds, Amer. J. Math., Volume 114 (1992) no. 3, pp. 495-550 | DOI | MR | Zbl

[21] Semmes, Stephen The homogeneous complex Monge-Ampère equation and the infinite-dimensional versions of classic symmetric spaces, The Gelfand Mathematical Seminars, 1993–1995 (Gelfand Math. Sem.), Birkhäuser Boston, Boston, MA, 1996, pp. 225-242 | MR | Zbl

[22] Shiffman, Bernard; Tate, Tatsuya; Zelditch, Steve Harmonic analysis on toric varieties, Explorations in complex and Riemannian geometry (Contemp. Math.), Volume 332, Amer. Math. Soc., Providence, RI, 2003, pp. 267-286 | MR | Zbl

[23] Shiffman, Bernard; Tate, Tatsuya; Zelditch, Steve Distribution laws for integrable eigenfunctions, Ann. Inst. Fourier (Grenoble), Volume 54 (2004) no. 5, pp. 1497-1546 | DOI | Numdam | MR | Zbl

[24] Song, Jian The α-invariant on toric Fano manifolds, Amer. J. Math., Volume 127 (2005) no. 6, pp. 1247-1259 | DOI | MR | Zbl

[25] Song, Jian; Zelditch, S. Bergman metrics and geodesics in the space of Kähler metrics on toric varieties (preprint, arXiv: 0707.3082)

[26] Tian, Gang On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom., Volume 32 (1990) no. 1, pp. 99-130 | MR | Zbl

[27] Yau, Shing Tung On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I, Comm. Pure Appl. Math., Volume 31 (1978) no. 3, pp. 339-411 | DOI | MR | Zbl

[28] Yau, Shing-Tung Open problems in geometry, Chern—a great geometer of the twentieth century, Int. Press, Hong Kong, 1992, pp. 275-319 | MR

[29] Zelditch, Steve Bernstein polynomials, Bergman kernels and toric Kähler varieties (preprint, arXiv: 0705.2879)

[30] Zelditch, Steve Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices (1998) no. 6, pp. 317-331 | DOI | MR | Zbl

Cité par Sources :