On singularities of generically immersive holomorphic maps between complex hyperbolic space forms

Abstract

In 1965, Feder proved using a cohomological identity that any holomorphic immersion t: Pn→Pm between complex projective spaces is necessarily a linear embedding whenever m < 2n. In 1991, Cao-Mok adapted Feder’s identity to study the dual situation of holomorphic immersions between compact complex hyperbolic space forms, proving that any holomorphic immersion f : X→Y from an n-dimensional compact complex hyperbolic space form X into any m-dimensional complex hyperbolic space form Y must necessarily be totally geodesic provided that m < 2n. We study in this article singularity loci of generically injective holomorphic immersions between complex hyperbolic space forms. Under dimension restrictions, we show that the open subset U over which the map is a holomorphic immersion cannot possibly contain compact complex-analytic sub-varieties of large dimensions which are in some sense sufficiently deformable. While in the finite-volume case it is enough to apply the arguments of Cao-Mok, the main input of the current article is to introduce a geometric argument that is completely local. Such a method applies to f: X→Y in which the complex hyperbolic space form X is possibly of infinite volume. To start with we make use of the Ahlfors-Schwarz Lemma, as motivated by recent work of Koziarz-Mok, and reduce the problem to the local study of contracting leafwise holomorphic maps between open subsets of complex unit balls. Rigidity results are then derived from a commutation formula on the complex Hessian of the holomorphic map.