Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

TL;DR

This paper investigates the structure of compact Kaehler manifolds with numerically effective tangent bundles that admit holomorphic Cartan geometries, proving the fibers are rational homogeneous and the associated principal bundle has a flat connection.

Contribution

It proves that fibers are rational homogeneous varieties and the principal G-bundle over the torus admits a flat holomorphic connection under the given conditions.

Findings

Fibers of the submersion are rational homogeneous varieties.

The principal G-bundle admits a flat holomorphic connection.

Supports the conjecture that fibers are rational and homogeneous.

Abstract

Let X be a compact connected Kaehler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly, Peternell and Schenider says that there is a finite unramified Galois covering M --> X, a complex torus T, and a holomorphic surjective submersion f: M --> T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.