Abstract
In Biswas and Dumitrescu (2018), we introduced and studied the concept of holomorphic 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 ℂPd-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).
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