Nonexistence of Proper Holomorphic Maps Between Certain Classical Bounded Symmetric Domains

The author,motivated by his results on Hermitian metric rigidity,conjectured in [4] 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.The Conjecture was resolved in the affirmative by I.-H.Tsai [8].When the hypothesis r′≤r is removed,the structure of proper holomorphic maps f：Ω→Ω′is far from being understood,and the complexity in studying such maps depends very much on the difference r′-r,which is called the rank defect.The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H.Tu [10],in which a rigidity theorem was proven for certain pairs of classical domains of type I,which implies nonexistence theorems for other pairs of such domains.For both results the rank defect is equal to 1,and a generaliza- tion of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of （Ω→Ω′） of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and L-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case.

Proper Holomorphic Maps from Domains in C2 with Transverse Circle Action

The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space,where the domain is of finite type and admits a transverse circle action.The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.

On the Proper Holomorphic Mappings between Equidimensional Hartogs Domains over Hermitian Symmetric Manifolds

In this paper,we study a family of Hartogs domains fibred over Hermitian symmetric manifolds being a unit ball in C^(m).The aim of the present study is to establish the rigidity results about proper holomorphic mappings between two equidimensional Hartogs domains over Hermitian symmetric manifolds.In particular,we can fully determine its biholomorphic equivalence and automorphism group.

Extension of Proper _-holom orphic Mappings

In this paper,we prove that a proper μ holomorphic mapping f:D 1→D 2 between bounded domains with some convexity,such that f satisfies some growth condition,extends smoothly to bD 1-{z:U(z)=0}.

Classification of Proper Holomorphic Mappings between Hartogs Domains over Homogeneous Siegel Domains

The Hartogs domain over homogeneous Siegel domain D_(N,s)(s>0)is defined by the inequality■,where D is a homogeneous Siegel domain of typeⅡ,(z,ζ)∈D×C~N and KD(z,z)is the Bergman kernel of D.Recently,Seo obtained the rigidity result that proper holomorphic mappings between two equidimensional domains D_(N,s)and D'_(N',s')are biholomorphisms for N≥2.In this article,we find a counter-example to show that the rigidity result is not true for D_(1,s)and obtain a classification of proper holomorphic mappings between D_(1,s)and D'_(1,s').

PROPER HOLOMORPHIC MAPPINGS BETWEEN SOME NONSMOOTH DOMAINS

The authors charaCterize the existence of proper holomorphic mapping between a kind of nonsmooth Reinhardt domains defined as

The Classification of Proper Holomorphic Mappings Between Special Hartogs Triangles of Different Dimensions

The authors discuss the proper holomorphic mappings between special Hartogs triangles of different dimensions and obtain a corresponding classification theorem.

Proper Holomorphic Mappings Between n-generalized Hartogs Triangles

In this paper,we first introduce the notion of n-generalized Hartogs triangles.Then,we characterize proper holomorphic mappings between some of these domains,and describe their automorphism groups.

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

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.