Normal Holonomy of Complex Hyperbolic Submanifolds

2025/06/12 by Santiago Castañeda Montoya, Montoya, Santiago Castañeda, Carlos Olmos +1
Mathematics · #Geometric and Algebraic Topology #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2506.11323

Abstract

We prove that the restricted normal holonomy group of a Kähler submanifold of the complex hyperbolic space ℂHⁿ is always transitive, provided the index of relative nullity is zero. This contrasts with the case of ℂPⁿ, where a Berger type result was proved by Console, Di Scala, and the second author. The proof is based on lifting the submanifold to the pseudo-Riemannian space ℂ^n,1 and developing new tools to handle the difficulties arising from possible degeneracies in holonomy tubes and associated distributions. In particular, we introduce the notion of weakly polar actions and a framework for dealing with degenerate submanifolds. These techniques could contribute to a broader understanding of submanifold geometry in spaces with indefinite signature, offering new insight into submanifolds in the dual setting of complex projective geometry.

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