Let and be two Kähler manifolds. We call and relatives if they share a non-trivial Kähler submanifold , namely, if there exist two holomorphic and isometric immersions (Kähler immersions) and . Moreover, two Kähler manifolds and are said to be weakly relatives if there exist two locally isometric (not necessarily holomorphic) Kähler manifolds and which admit two Kähler immersions into and respectively. The notions introduced are not equivalent (cf. Example 2.3). Our main results in this paper are Theorem 1.2 and Theorem 1.4. In the first theorem we show that a complex bounded domain with its Bergman metric and a projective Kähler manifold (i.e. a projective manifold endowed with the restriction of the Fubini–Study metric) are not relatives. In the second theorem we prove that a Hermitian symmetric space of noncompact type and a projective Kähler manifold are not weakly relatives. Notice that the proof of the second result does not follows trivially from the first one. We also remark that the above results are of local nature, i.e. no assumptions are used about the compactness or completeness of the manifolds involved.
@article{ASNSP_2010_5_9_3_495_0, author = {Di Scala, Antonio and Loi, Andrea}, title = {K\"ahler manifolds and their relatives}, journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze}, pages = {495--501}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 9}, number = {3}, year = {2010}, mrnumber = {2722652}, zbl = {1253.53066}, language = {en}, url = {http://archive.numdam.org/item/ASNSP_2010_5_9_3_495_0/} }
TY - JOUR AU - Di Scala, Antonio AU - Loi, Andrea TI - Kähler manifolds and their relatives JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2010 SP - 495 EP - 501 VL - 9 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://archive.numdam.org/item/ASNSP_2010_5_9_3_495_0/ LA - en ID - ASNSP_2010_5_9_3_495_0 ER -
%0 Journal Article %A Di Scala, Antonio %A Loi, Andrea %T Kähler manifolds and their relatives %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2010 %P 495-501 %V 9 %N 3 %I Scuola Normale Superiore, Pisa %U http://archive.numdam.org/item/ASNSP_2010_5_9_3_495_0/ %G en %F ASNSP_2010_5_9_3_495_0
Di Scala, Antonio; Loi, Andrea. Kähler manifolds and their relatives. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 3, pp. 495-501. http://archive.numdam.org/item/ASNSP_2010_5_9_3_495_0/
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