The normal holonomy group of Kähler submanifolds

Abstract

We study the (restricted) holonomy group ${\rm Hol}(\nabla^{\perp})$ of the normal connection $\nabla^{\perp}$ (shortened to normal holonomy group) of a Kähler submanifold of a complex space form. We prove that if the normal holonomy group acts irreducibly on the normal space then it is linear isomorphic to the holonomy group of an irreducible Hermitian symmetric space. In particular, it is a compact group and the complex structure $J$ belongs to its Lie algebra.

We prove that the normal holonomy group acts irreducibly if the submanifold is full (that is, it is not contained in a totally geodesic proper Kähler submanifold) and the second fundamental form at some point has no kernel. For example, a Kähler–Einstein submanifold of $\mathbb{C} P^n$ has this property.

We define a new invariant $\mu$ of a Kähler submanifold of a complex space form. For non-full submanifolds, the invariant $\mu$ measures the deviation of $J$ from belonging to the normal holonomy algebra. For a Kähler–Einstein submanifold, the invariant $\mu$ is a rational function of the Einstein constant. By using the invariant $\mu$, we prove that the normal holonomy group of a not necessarily full Kähler–Einstein submanifold of $\mathbb{C} P^n$ is compact, and we give a list of possible holonomy groups.

The approach is based on a definition of the holonomy algebra ${\rm hol}(P)$ of an arbitrary curvature tensor field $P$ on a vector bundle with a connection and on a De Rham type decomposition theorem for ${\rm hol}(P)$.