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. In particular, this answers a question posed by Mok twenty years ago about decomposable normal bundles for complex surfaces immersed in four-dimensional complex hyperbolic space forms. The proof uses the polarized complex variation of Hodge structure induced from the target. In this talk, we mainly introduce the background of the problem, the basic knowledge of complex hyperbolic geometry, and simple ideas of proof.