2011 | OriginalPaper | Buchkapitel
On singularities of generically immersive holomorphic maps between complex hyperbolic space forms
verfasst von : Ngaiming Mok
Erschienen in: Complex and Differential Geometry
Verlag: Springer Berlin Heidelberg
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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.