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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access February 8, 2017

Strongly not relatives Kähler manifolds

Michela Zedda EMAIL logo
From the journal Complex Manifolds


In this paper we study Kähler manifolds that are strongly not relative to any projective Kähler manifold, i.e. those Kähler manifolds that do not share a Kähler submanifold with any projective Kähler manifold even when their metric is rescaled by the multiplication by a positive constant. We prove two results which highlight some relations between this property and the existence of a full Kähler immersion into the infinite dimensional complex projective space. As application we get that the 1-parameter families of Bergman-Hartogs and Fock-Bargmann-Hartogs domains are strongly not relative to projective Kähler manifolds.


[1] E. Bi, Z. Feng, Z. Tu, Balanced metrics on the Fock-Bargmann-Hartogs domains, Ann Glob Anal Geom 49 (2016), n. 4, 349-359.Search in Google Scholar

[2] S. Bochner, Curvature in Hermitian metric, Bull. Amer. Math. Soc. 53 (1947), 179-195.10.1090/S0002-9904-1947-08778-4Search in Google Scholar

[3] E. Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1-23.10.2307/1969817Search in Google Scholar

[4] X. Cheng, A. J. Di Scala, Y. Yuan, Kähler submanifolds and the Umehara Algebra, preprint 2016, arXiv: 1601.05907v2 [math.DG].Search in Google Scholar

[5] A. J. Di Scala, H. Ishi, A. Loi, Kähler immersions of homogeneous Kähler manifolds into complex space forms, Asian J. Math. (3) 16 (2012), 479-488.10.4310/AJM.2012.v16.n3.a7Search in Google Scholar

[6] A. J. Di Scala, A. Loi, Kähler manifolds and their relatives, Ann. Scuola Normale Pisa (3) 9 (2010), 495-501.Search in Google Scholar

[7] Z. Feng, Z. Tu, Balanced metrics on some Hartogs type domains over bounded symmetric domains, Ann. of Glob. Anal. Geom. 47 (2015), (4), 305-333.10.1007/s10455-014-9447-8Search in Google Scholar

[8] Y. Hao, A. Wang, Kähler geometry of bounded pseudoconvex Hartogs domains, preprint 2014, arXiv:1411.4447 [math.CV].Search in Google Scholar

[9] Y. Hao, A.Wang, The Bergman kernels of generalized Bergman-Hartogs domains, J.Math. Anal. Appl. 429 (2015), 1, 326-336.10.1016/j.jmaa.2015.04.023Search in Google Scholar

[10] X. Huang, Y. Yuan, Submanifolds of Hermitian symmetric spaces, Springer Proc. Math. Stat. 127 (2015), Analysis and Geometry, Springer, 197-206.10.1007/978-3-319-17443-3_10Search in Google Scholar

[11] S. Kobayashi, Geometry of Bounded Domains, Trans. Amer. Math. Soc. 92 (1996), 267-290.10.1090/S0002-9947-1959-0112162-5Search in Google Scholar

[12] A. Loi, R. Mossa, Berezin quantization of homogeneous bounded domains , Geom. Ded. 161 (2012), 1, 119-128.10.1007/s10711-012-9697-1Search in Google Scholar

[13] A. Loi, M. Zedda, Kähler-Einstein submanifolds of the infinite dimensional projective space,Math. Ann. 350 (2011), 145-154.10.1007/s00208-010-0554-ySearch in Google Scholar

[14] A. Loi, M. Zedda, Balanced metrics on Cartan and Cartan-Hartogs domains, Math. Zeit. 270 (3-4), 1077-1087.10.1007/s00209-011-0842-6Search in Google Scholar

[15] R. Mossa, A bounded homogeneous domain and a projective manifold are not relatives. Riv. Mat. Univ. Parma 4 (2013), (1), 55-59.Search in Google Scholar

[16] M. Umehara, Kähler submanifolds of complex space forms, Tokyo J. Math. 10 (1987), 1, 203-214.10.3836/tjm/1270141804Search in Google Scholar

[17] M. Zedda, Berezin-Engliš’ quantization of Cartan-Hartogs domains, J. Geom. Phys. 100 (2016), 62-67.10.1016/j.geomphys.2015.11.002Search in Google Scholar

[18] J. Zhao, A.Wang, Y. Hao, On the holomorphic automorphism group of the Bergman-Hartogsdomain, Int. J.Math. (8) 26 (2015), art. n. 1550056.Search in Google Scholar

Received: 2016-9-24
Accepted: 2016-11-6
Published Online: 2017-2-8
Published in Print: 2017-2-23

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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