Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents
type: Item
instance of: scholarly article
title: Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents
zbMATH Open document ID: 1165.32008
author: Ngaiming Mok
published in: Science in China. Series A
publication date: 2008-09-09T00:00:00Z
review text: Motivated by the relations between function theory on bounded symmetric domains and the geometric theory of Fano manifolds, the author gives a new proof of a theorem of Tsai in the case of irreducible symmetric domains of type I. This theorem states that a proper holomorphic map between an irreducible bounded symmetric domain of rank \(r>1\) and another bounded domain of rank \(r' \leq r\) is necessarily totally geodesic. The proof is in two steps. First, the author embeds the domain in the Grassmann compactification and uses boundary values to prove that the map \( f\) transforms rank 1 vectors to rank 1 vectors, on a nonempty open set. Then, thanks to the study of geometric structures by means of rational curves which provides a new proof of a result of Neretin, he proves that \(f\) is actually a standard embedding between complex Grassmann manifolds.
reviewed by: Erwan Rousseau
Mathematics Subject Classification ID: 32H35
Mathematics Subject Classification ID: 14J45
Mathematics Subject Classification ID: 32M15
zbMATH DE Number: 5323346
zbMATH Keywords: minimal rational curves
zbMATH Keywords: varieties of minimal rational tangents
zbMATH Keywords: analytic continuation
zbMATH Keywords: rigidity
zbMATH Keywords: bounded symmetric domains
zbMATH Keywords: proper holomorphic maps
MaRDI profile type: Publication
full work available at URL: s11425-008-0005-3
OpenAlex ID: W1986809194
cites work: Holomorphic maps from rational homogeneous spaces of Picard number 1 onto projective manifolds
cites work: Q3992565
cites work: Rigidity of proper holomorphic maps between symmetric domains
cites work: Q3993815
cites work: Q4949409
cites work: Cartan-Fubini type extension of holomorphic maps for Fano manifolds of Picard number 1.
cites work: Birationality of the tangent map for minimal rational curves
cites work: Conformal geometry of symmetric spaces and generalized linear-fractional maps of Krein-Shmul'yan
cites work: Rigidity of smooth Schubert varieties in Hermitian symmetric spaces
cites work: Q5673183
cites work: Geometry Associated with Semisimple Flat Homogeneous Spaces
cites work: Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation
cites work: Q5690067
cites work: Families of singular rational curves
cites work: Q4409552
cites work: The ramification divisors for branched coverings of \(P_ k^ n\)
cites work: Q3140224
cites work: Q4225283
cites work: Integral varieties of the canonical cone structure on \(G/P\)
cites work: Uniruled projective manifolds with irreducible reductive G-structures.
cites work: Rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains
cites work: On Stein extensions of real symmetric spaces
cites work: Q3769563
DOI: 10.1007/S11425-008-0005-3
description: scientific article; zbMATH DE number 5323346