Analytic continuation on bounded symmetric domains and uniruled projective manifolds

Abstract

Analytic continuation is a central issue in Several Complex Variables, starting with theHartogs Phenomenon. We examine techniques of analytic continuation for irreducible boundedsymmetric domains Ω and their dual Hermitian symmetric spaces of the compact typeS, andtheir generalizations to uniruled projective manifolds.As a starting point, for rank(S)≥2 we recall a proof using local differential geometry andthe Hartogs Phenomenon of a theorem of Ochiai (1970) for the analytic continuation of flatS-structures, and its generalization to the Cartan-Fubini extension principle of Hwang-Mok(2001) in the geometric theory of uniruled projective manifolds modeled on varieties of minimalrational tangents (VMRTs). Applying CR-geometry, Mok-Ng (2012) proved that under anondegeneracy assumption, a germ of measure-preserving holomorphic mapf: (Ω,λdμΩ; 0)→(Ω,dμΩ; 0)×···×(Ω,dμΩ; 0), wheredμΩdenotes the Bergman volume form andλ >0 is a realconstant, is necessarily a totally geodesic embedding, answering in the affirmative a questionof Clozel-Ullmo (2003) regarding commutants of Hecke correspondences. The proof involves anew Alexander-type extension theorem for irreducible bounded symmetric domains Ω of rank≥2.In another direction we explain the non-equidimensional Cartan-Fubini extension princi-ple of Hong-Mok (2010). We consider furthermore the problem of analytic continuation ofsubvarieties of uniruled projective manifolds (X,K) equipped with a VMRT-structure underthe assumption that the subvariety inherits a sub-VMRT structure defined by intersections ofVMRTs with projectivized tangent spaces, and establish a principle of analytic continuation(Mok-Zhang 2015) under auxiliary conditions by constructing a universal family of chains ofrational curves by an analytic process and proving its algebraicity by establishing a Thullenextension theorem on a paramentrized family of sub-VMRT structures along chains of rationalcurves.