From transcendence to algebraicity: techniques of analytic continuation on bounded symmetric domains and their dual compact Hermitian symmetric spaces

Abstract

Analytic continuation is a central issue in Several Complex Vari-ables, starting with the Hartogs Phenomenon. We examine the applicationsof techniques of analytic continuation in Complex Geometry for irreduciblebounded symmetric domains Ω and their dual Hermitian symmetric spaces ofthe compact typeS, and their ramifications to the geometric theory of unir-uled projective manifolds. As a starting point, in the case where rank(S)≥2 we recall a proof of Ochiai’s theorem (1970) for analytic continuation of flatS-structure using Hartogs extension, and its generalization to the Cartan-Fubini extension principle of Hwang-Mok (2001) in the geometric theory ofuniruled projective manifolds basing on varieties of minimal rational tangents(VMRTs). Applying methods of algebraic extension in CR-geometry of Web-ster and Huang, and Ochiai’s theorem, we give the proof of Mok-Ng (2012)that under a nondegeneracy assumption, a germ of measure-preserving holo-morphic mapf: (Ω,λdμΩ; 0)→(Ω,dμΩ; 0)×···×(Ω,dμΩ; 0), wheredμΩdenotes the Bergman volume form andλ >0 is a real constant, is necessarilya totally geodesic diagonal embedding, answering in the affirmative a problemof Clozel-Ullmo stemming from a problem in Arithmetic Dynamics regard-ing Hecke correspondences. The proof involves Alexander’s Theorem for thecomplex unit ballBn,n≥2, in the rank-1 case and a new Alexander-type ex-tension theorem for the case of irreducible bounded symmetric domains Ω ofrank≥2 for germs of holomorphic maps preserving the regular part Reg(∂Ω)of the boundary. In another direction we explain the non-equidimensionalCartan-Fubini extension principle of Hong-Mok (2010) and its application tothe characterization of smooth Schubert varieties in rational homogeneousmanifolds of Picard number 1 (Hong-Mok 2013). Finally, we consider theproblem of analytic continuation of subvarieties of uniruled projective man-ifolds (X,K) equipped with a VMRT-structure (e.g. irreducible Hermitiansymmetric spacesSof the compact type) under the assumption that thesubvariety inherits a sub-VMRT structure by taking intersections of VMRTswith tangent spaces, and establish a principle of analytic continuation (Mok-Zhang 2015) by a parametrized Thullen extension of sub-VMRT structuresalong chains of rational curves.