Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents  

作　　者：MOK Ngaiming 

机构地区：[1]Department of Mathematics,The University of Hong Kong

出　　处：《Science China Mathematics》2008年第4期660-684,共25页中国科学：数学（英文版）

基　　金：This research is partially supported by a Competitive Earmarked Research Grant of the Research Grants Council of Hong Kong,China

摘　　要：In 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ? 2 into a bounded symmetric domain Ω′ is necessarily totally geodesic provided that r′:= rank(Ωg’) ? rank(Ω):= r, proving a conjecture of the author’s motivated by Hermitian metric rigidity. As a first step in the proof, Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1. Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding, this means that the germ of f at a general point preserves the varieties of minimal rational tangents (VMRTs).In another completely different direction Hwang-Mok established with very few exceptions the Cartan-Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard number 1, showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs. We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1, especially in the case of classical manifolds such as rational homogeneous spaces of Picard number 1, by a non-equidimensional analogue of the Cartan-Fubini extension principle. As an illustration we show along this line that standard embeddings between complex Grassmann manifolds of rank ? 2 can be characterized by the VMRT-preserving property and a non-degeneracy condition, giving a new proof of a result of Neretin’s which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1, on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω’ from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ’ is necessarily totally geodesic provided that r’:=rank(Ω’)≤rank(Ω):= r,proving a conjecture of the author’s motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin’s which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.

关 键 词：minimal rational curves varieties of minimal rational tangents analytic continuation RIGIDITY bounded symmetric domains proper holomorphic maps 14J45 32M15 32H35 53C10 

分 类 号：O174.5[理学—数学]