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

In 1965, Feder proved using a cohomological identity that any holomorphic immersion

$$\tau : {\mathbb{P}^n} \rightarrow {\mathbb{P}^m}$$

between complex projective spaces is necessarily a linear embedding whenever

m

< 2

n

. 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

$$ f : X \rightarrow Y $$

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

< 2

n

. 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 subvarieties of large dimensions which are in some sense sufficiently deformable. While in the finitevolume 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

$$ f : X \rightarrow Y $$

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.