Les domaines de Cartan–Hartogs sont définis comme une classe de domaines de type Hartogs sur les domaines symétriques bornés irréductibles. Pour un domaine de Cartan–Hartogs muni de sa métrique de Kähler naturelle , Zedda a conjecturé que le coefficient du développement de la fonction ε de Rawnsley relative au domaine de Cartan–Hartogs est constant sur si et seulement si est biholomorphiquement isométrique à l'espace hyperbolique complexe. Dans cet article, en nous appuyant sur ses arguments, nous donnons une preuve géométrique de la conjecture de Zedda en calculant les tenseurs de courbure du domaine de Cartan–Hartogs .
The Cartan–Hartogs domains are defined as a class of Hartogs-type domains over irreducible bounded symmetric domains. For a Cartan–Hartogs domain endowed with the natural Kähler metric , Zedda conjectured that the coefficient of the Rawnsley's ε-function expansion for the Cartan–Hartogs domain is constant on if and only if is biholomorphically isometric to the complex hyperbolic space. In this paper, following Zedda's argument, we give a geometric proof of the Zedda's conjecture by computing the curvature tensors of the Cartan–Hartogs domain .
Accepté le :
Publié le :
@article{CRMATH_2017__355_7_760_0, author = {Bi, Enchao and Tu, Zhenhan}, title = {Remarks on the canonical metrics on the {Cartan{\textendash}Hartogs} domains}, journal = {Comptes Rendus. Math\'ematique}, pages = {760--768}, publisher = {Elsevier}, volume = {355}, number = {7}, year = {2017}, doi = {10.1016/j.crma.2017.06.009}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2017.06.009/} }
TY - JOUR AU - Bi, Enchao AU - Tu, Zhenhan TI - Remarks on the canonical metrics on the Cartan–Hartogs domains JO - Comptes Rendus. Mathématique PY - 2017 SP - 760 EP - 768 VL - 355 IS - 7 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2017.06.009/ DO - 10.1016/j.crma.2017.06.009 LA - en ID - CRMATH_2017__355_7_760_0 ER -
%0 Journal Article %A Bi, Enchao %A Tu, Zhenhan %T Remarks on the canonical metrics on the Cartan–Hartogs domains %J Comptes Rendus. Mathématique %D 2017 %P 760-768 %V 355 %N 7 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2017.06.009/ %R 10.1016/j.crma.2017.06.009 %G en %F CRMATH_2017__355_7_760_0
Bi, Enchao; Tu, Zhenhan. Remarks on the canonical metrics on the Cartan–Hartogs domains. Comptes Rendus. Mathématique, Tome 355 (2017) no. 7, pp. 760-768. doi : 10.1016/j.crma.2017.06.009. http://www.numdam.org/articles/10.1016/j.crma.2017.06.009/
[1] On the curvature tensor of the Hermitian symmetric manifolds, Ann. of Math. (2), Volume 71 (1960) no. 3, pp. 508-521
[2] Quantization of Kähler manifolds. I: geometric interpretation of Berezin's quantization, J. Geom. Phys., Volume 7 (1990), pp. 45-62
[3] On compact, locally symmetric Kähler manifolds, Ann. of Math., Volume 71 (1960) no. 3, pp. 472-507
[4] The Bergman kernel and a theorem of Tian, Katata, 1997 (Trends in Mathematics), Birkhäuser Boston, Boston, MA (1999), pp. 1-23
[5] Scalar curvature and projective embeddings, I, J. Differ. Geom., Volume 59 (2001), pp. 479-522
[6] A Forelli–Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Anal., Volume 177 (2000) no. 2, pp. 257-281
[7] The asymptotics of a Laplace integral on a Kähler manifold, J. Reine Angew. Math., Volume 528 (2000), pp. 1-39
[8] On canonical metrics on Cartan–Hartogs domains, Math. Z., Volume 278 (2014) no. 1, pp. 301-320
[9] Differential Geometry, Lie groups, and Symmetric Spaces, Academic Press, 1979
[10] Analytic invariants of bounded symmetric domains, Proc. Amer. Math. Soc., Volume 19 (1968) no. 2, pp. 279-284
[11] Balanced metrics on Cartan and Cartan–Hartogs domains, Math. Z., Volume 270 (2012), pp. 1077-1087
[12] On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch, Amer. J. Math., Volume 122 (2000) no. 2, pp. 235-273
[13] Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics, vol. 254, Birkhäuser Boston Inc., Boston, MA, USA, 2007
[14] Generalized Bergman kernels on symplectic manifolds, Adv. Math., Volume 217 (2008) no. 4, pp. 1756-1815
[15] Berezin–Toeplitz quantization on Kähler manifolds, J. Reine Angew. Math., Volume 662 (2012), pp. 1-56
[16] Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds, World Scientific, 1989
[17] Coherent states and Kähler manifolds, Q. J. Math., Volume 28 (1977) no. 2, pp. 403-415
[18] Strong rigidity of compact quotients of exceptional bounded symmetric domains, Duke Math. J., Volume 48 (1981) no. 4, pp. 857-871
[19] Lie Groups, Lie Algebras, and Their Representation, Springer, 1984
[20] A closed formula for the asymptotic expansion of the Bergman kernel, Commun. Math. Phys., Volume 314 (2012), pp. 555-585
[21] The equivalence on classical metrics, Sci. China Ser. A, Math., Volume 50 (2007) no. 2, pp. 183-200
[22] Canonical metrics on Cartan–Hartogs domains, Int. J. Geom. Methods Mod. Phys., Volume 09 (2012) no. 1
[23] Szegö kernels and a theorem of Tian, Int. Math. Res. Not., Volume 6 (1998), pp. 317-331
Cité par Sources :