Split Normal Bundles and Rigidity of Holomorphic Immersions from Compact Ball Quotients

Abstract

Let $f$ be a holomorphic immersion from a compact complex hyperbolic space form into a higher-dimensional complex hyperbolic space form. We prove that if the normal bundle decomposes as a direct sum of holomorphic line bundles, then the immersion is totally geodesic. This result has no restriction on the codimension and extends, under the splitting hypothesis, the classical rigidity theorem of Cao--Mok. The proof uses the polarized complex variation of Hodge structure induced from the target. The line-bundle decomposition produces nef zero-Higgs line quotients, and a Chern-character identity forces the Gauss-defect form to have nontrivial nullity at every point. The null-foliation and boundary argument of Cao--Mok then shows that the defect vanishes identically. In particular, this answers a question posed by Mok almost twenty years ago about decomposable normal bundles for complex surfaces immersed in four-dimensional complex hyperbolic space forms.