Abstract
This paper consists of two results dealing with balanced metrics (in Donaldson terminology) on noncompact complex manifolds. In the first one we describe all balanced metrics on Cartan domains. In the second one we show that the only Cartan–Hartogs domain which admits a balanced metric is the complex hyperbolic space. By combining these results with those obtained in Loi and Zedda (Mathematische Annalen, 2011, to appear) we also provide the first example of complete, Kähler-Einstein and projectively induced metric g such that α g is not balanced for all α > 0.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arazy J.: A survey of invariant Hilbert spaces of analytic functions on bounded symmetric domains, Contemp. Math. 185, 7–65 (1995)
Arezzo C., Loi A.: Moment maps, scalar curvature and quantization of Kähler manifolds. Comm. Math. Phys. 243, 543–559 (2004)
Cahen M., Gutt S., Rawnsley J.: Quantization of Kähler manifolds. I. Geometric interpretation of Berezin’s quantization. J. Geom. Phys. 7, 45–62 (1990)
Calabi E.: Isometric imbeddings of complex manifolds. Ann. Math. 58, 1–23 (1953)
Cuccu F., Loi A.: Global symplectic coordinates on complex domains. J. Geom. Phys. 56, 247–259 (2006)
Donaldson S.: Scalar curvature and projective embeddings, I. J. Diff. Geom. 59, 479–522 (2001)
Engliš M.: Weighted Bergman kernels and balanced metrics. RIMS Kokyuroku 1487, 40–54 (2006)
Faraut J., Koranyi A.: Function spaces and reproducing kernels on bounded symmetric domains. J. Funct. Anal. 88, 64–89 (1990)
Gramchev T., Loi A.: TYZ expansion for the Kepler manifold. Comm. Math. Phys. 289, 825–840 (2009)
Greco A., Loi A.: Radial balanced metrics on the unit disk. J. Geom. Phys. 60, 53–59 (2010)
Loi A.: Regular quantizations of Kähler manifolds and constant scalar curvature metrics. J. Geom. Phys. 53, 354–364 (2005)
Loi, A., Zedda, M.: Kähler–Einstein submanifolds of the infinite dimensional projective space, Mathematische Annalen (2011, to appear)
Roos G., Lu K., Yin W.: New classes of domains with explicit Bergman kernel. Sci. China 47(3), 352–371 (2004)
Roos, G., Wang, A., Yin, W. P., Zhang, L. Y.: The Kähler-Einstein metric for some Hartogs domains over bounded symmetric domains, Science in China 49, 1175–1210 (September 2006)
Rawnsley J.: Coherent states and Kähler manifolds, Quart. Quart. J. Math. Oxford (2), 28, 403–415 (1977)
Upmeier, H.: Index theory for Toeplitz operators on bounded symmetric domains. In: Représentations des groupes et analyse complexe (Luminy, 1986),vol. 24, pp. 89–94, Journées SMF, Univ. Poitiers, Poitiers (1986)
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by the M.I.U.R. Project “Geometric Properties of Real and Complex Manifolds”; the second author was supported by RAS through a grant financed with the “Sardinia PO FSE 2007–2013” funds and provided according to the L.R. 7/2007.
Rights and permissions
About this article
Cite this article
Loi, A., Zedda, M. Balanced metrics on Cartan and Cartan–Hartogs domains. Math. Z. 270, 1077–1087 (2012). https://doi.org/10.1007/s00209-011-0842-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00209-011-0842-6