math.DG

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# Title: Transversely holomorphic branched Cartan geometry

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).
 Subjects: Differential Geometry (math.DG); Complex Variables (math.CV) MSC classes: 53C05, 53C12, 55R55 Cite as: arXiv:1803.06472 [math.DG] (or arXiv:1803.06472v1 [math.DG] for this version)

## Submission history

From: Indranil Biswas [view email]
[v1] Sat, 17 Mar 2018 06:16:43 GMT (13kb)