On the structure of complete k\"ahlerian manifolds furnished with closed conformal vector fields

TL;DR

This paper characterizes the structure of complete Kähler manifolds with closed conformal vector fields, showing under certain curvature and holonomy conditions that such manifolds are foliated by totally geodesic tori and other Kähler submanifolds, often reducing to flat or product geometries.

Contribution

It provides new rigidity results for complete Kähler manifolds with closed conformal vector fields, extending known classifications under curvature and holonomy assumptions.

Findings

A compact Kähler surface with nonpositive curvature and a closed conformal vector field is a flat torus with a parallel vector field.

Under certain conditions, a complete Kähler manifold admits a foliation by totally geodesic tori and lower-dimensional Kähler manifolds.

The universal cover of such a manifold is isometric to a Riemannian product with an  factor.

Abstract

We show that if a connected compact k\"ahlerian surface $M$ with nonpositive gaussian curvature is furnished with a closed conformal vector field $\xi$ whose singular points are isolated, then $M$ is isometric to a flat torus and $\xi$ is parallel. We also consider the case of a connected complete k\"ahlerian manifod $M$ of complex dimension $n>1$ and furnished with a nontrivial closed conformal vector field $\xi$. In this case, it is well known that the singularities of $\xi$ are automatically isolated and the nontrivial leaves of the distribution generated by $\xi$ and $J\xi$ are totally geodesic in $M$. Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of $M$ along it is nonpositive, we show that $\xi$ is parallel and $M$ is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete k\"ahlerian manifolds of complex dimension $n-1$. In particular, the the universal covering of $M$ is isometric to a riemannian product having $\mathbb R^2$ as a factor. We also present a generic example showing that one cannot get rid of the hypothesis on the nonpositivity of the holomorphic sectional curvature along at least one such leaf.