ASYMPTOTIC TOTAL GEODESY OF LOCAL HOLOMORPHIC CURVES EXITING A BOUNDED SYMMETRIC DOMAIN AND APPLICATIONS TO A UNIFORMIZATION PROBLEM FOR ALGEBRAIC SUBSETS

CSpace

	ASYMPTOTIC TOTAL GEODESY OF LOCAL HOLOMORPHIC CURVES EXITING A BOUNDED SYMMETRIC DOMAIN AND APPLICATIONS TO A UNIFORMIZATION PROBLEM FOR ALGEBRAIC SUBSETS
	Chan, Shan Tai 1; Mok, Ngaiming
	2022
发表期刊	JOURNAL OF DIFFERENTIAL GEOMETRY
ISSN	0022-040X
卷号	120 期号:1 页码:1-49
摘要	The current article stems from our study on the asymptotic behavior of holomorphic isometric embeddings of the Poincare disk into bounded symmetric domains. As a first result we prove that any holomorphic curve exiting the boundary of a bounded symmetric domain Omega must necessarily be asymptotically totally geodesic. Assuming otherwise we derive by the method of rescaling a hypothetical holomorphic isometric embedding of the Poincare disk with Aut(Omega')-equivalent tangent spaces into a tube domain Omega' subset of Omega and derive a contradiction by means of the Poincare-Lelong equation. We deduce that equivariant holomorphic embeddings between bounded symmetric domains must be totally geodesic. Furthermore, we solve a uniformization problem on algebraic subsets Z subset of Omega. More precisely, if (Gamma) over cap subset of Aut(Omega) is a torsion-free discrete subgroup leaving Z invariant such that Z=(Gamma) over cap is compact, we prove that Z subset of Omega is totally geodesic. In particular, letting Gamma subset of Aut(Omega) be a torsion-free cocompact lattice, and pi : Omega -> Omega/Gamma =: X-Gamma be the uniformization map, a subvariety Y subset of X-Gamma must be totally geodesic whenever some (and hence any) irreducible component Z of pi(-1) (Y) is an algebraic subset of Omega. For cocompact lattices this yields a characterization of totally geodesic subsets of X-Gamma by means of bi-algebraicity without recourse to the celebrated monodromy result of Andre-Deligne on subvarieties of Shimura varieties, and as such our proof applies to not necessarily arithmetic cocompact lattices. In place of the monodromy result of Andre-Deligne we exploit the existence theorem of Aubin and Yau on Kahler-Einstein metrics for projective manifolds Y satisfying c(1)(Y) < 0 and make use of Nadel's semisimplicity theorem on automorphism groups of noncompact Galois covers of such manifolds, together with the total geodesy of equivariant holomorphic isometric embeddings between bounded symmetric domains.
收录类别	SCI
语种	英语
WOS研究方向	Mathematics
WOS类目	Mathematics
WOS记录号	WOS:000743374700001
出版者	INT PRESS BOSTON, INC
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/59887
专题	中国科学院数学与系统科学研究院
通讯作者	Chan, Shan Tai
作者单位	1.Chinese Acad Sci, Acad Math & Syst Sci, Inst Math, Beijing 100190, Peoples R China 2.Univ Hong Kong, Dept Math, Pokfulam Rd, Hong Kong, Peoples R China
推荐引用方式 GB/T 7714	Chan, Shan Tai,Mok, Ngaiming. ASYMPTOTIC TOTAL GEODESY OF LOCAL HOLOMORPHIC CURVES EXITING A BOUNDED SYMMETRIC DOMAIN AND APPLICATIONS TO A UNIFORMIZATION PROBLEM FOR ALGEBRAIC SUBSETS[J]. JOURNAL OF DIFFERENTIAL GEOMETRY,2022,120(1):1-49.
APA	Chan, Shan Tai,&Mok, Ngaiming.(2022).ASYMPTOTIC TOTAL GEODESY OF LOCAL HOLOMORPHIC CURVES EXITING A BOUNDED SYMMETRIC DOMAIN AND APPLICATIONS TO A UNIFORMIZATION PROBLEM FOR ALGEBRAIC SUBSETS.JOURNAL OF DIFFERENTIAL GEOMETRY,120(1),1-49.
MLA	Chan, Shan Tai,et al."ASYMPTOTIC TOTAL GEODESY OF LOCAL HOLOMORPHIC CURVES EXITING A BOUNDED SYMMETRIC DOMAIN AND APPLICATIONS TO A UNIFORMIZATION PROBLEM FOR ALGEBRAIC SUBSETS".JOURNAL OF DIFFERENTIAL GEOMETRY 120.1(2022):1-49.