Normal Holonomy of Complex Hyperbolic Submanifolds

TL;DR

This paper proves that the restricted normal holonomy group of a Kähler submanifold in complex hyperbolic space is always transitive when the index of nullity is zero, contrasting with known results in complex projective space.

Contribution

It introduces new tools and the notion of weakly polar actions to analyze degeneracies in holonomy tubes, advancing the understanding of submanifold geometry in indefinite signature spaces.

Findings

Normal holonomy is transitive for zero nullity in complex hyperbolic space.

Develops new methods to handle degeneracies in holonomy tubes.

Provides insights into submanifold geometry in indefinite signature spaces.

Abstract

We prove that the restricted normal holonomy group of a K\"ahler submanifold of the complex hyperbolic space $\mathbb{C}H^{n}$ is always transitive, provided the index of relative nullity is zero. This contrasts with the case of $\mathbb{C}P^{n}$, 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 $\mathbb{C}^{n,1}$ 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.