Transversely holomorphic branched Cartan geometry

TL;DR

This paper introduces a foliated version of transversely holomorphic branched Cartan geometry, demonstrating existence on complex manifolds of certain algebraic dimensions and flatness on specific classes of compact complex varieties.

Contribution

It defines a foliated version of branched Cartan geometry using Atiyah bundles and proves existence and flatness results on various complex manifolds.

Findings

Existence of transversely flat branched complex projective geometry on manifolds of algebraic dimension d.

Flatness of such geometries on rationally connected and Calabi-Yau manifolds.

Holomorphic maps into homogeneous spaces characterize these geometries.

Abstract

Earlier we introduced and studied the concept of holomorphic {\it branched Cartan geometry}. We define here a foliated version of this notion; this is done in terms of Atiyah bundle. We show that any complex compact manifold of algebraic dimension $d$ admits, away from a closed analytic subset of positive codimension, a nonsingular holomorphic foliation of complex codimension $d$ endowed with a transversely flat branched complex projective geometry (equivalently, a ${\mathbb C}P^d$-geometry). We also prove that transversely branched holomorphic Cartan geometries on compact complex projective rationally connected varieties and on compact simply connected Calabi-Yau manifolds are always flat (consequently, they are defined by holomorphic maps into homogeneous spaces).