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This is a survey paper dealing with holomorphic G-structures and holomorphic Cartan geometries on compact complex manifolds. Our emphasis is on the foliated case: holomorphic foliations with transverse (branched or generalized) holomorphic…
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…
In [DM] it was asked whether all flat holomorphic Cartan geometries (G,H) on a complex torus are translation invariant. We answer this affimatively under the assumption that the complex Lie group G is affine. More precisely, we show that…
In this paper we study transversely holomorphic foliations of complex codimension one with some hypothesis on the transverse structure.
We study codimension one holomorphic foliations on complex projective spaces and compact manifolds under the assumption that the foliation has a projective transverse structure in the complement of some invariant codimension one analytic…
We classify holomorphic Cartan geometries on every compact complex curve, and on every compact complex surface which contains a rational curve.
For a compact complex manifold, we introduce holomorphic foliations associated with certain abelian subgroups of the automorphism group. Such foliations are generalizations of holomorphic principal torus bundles. If there exists a…
This article studies codimension one foliations on projective man-ifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holo-nomy representation…
We study holomorphic foliations with an affine homogeneous transverse structure. We give a friendly characterization of the case of transversely affine foliations in terms of matrix valued pairs of differential forms. This leads naturally…
The category of exploded torus fibrations is an extension of the category of smooth manifolds in which some adiabatic limits look smooth. (For example, the limits considered in tropical geometry appear smooth, also degenerations…
We study holomorphic foliations of aribitrary codimension in smooth complete toric varieties. We show that split foliations are stable if some good behaviour of their singular set is provided. As an application of these results, we exhibit…
We study smooth foliations of arbitrary codimension on homogeneous compact K\"ahler manifolds. We prove that smooth foliations on rational compact homogeneous manifolds are locally trivial fibrations and classify the smooth foliations with…
A transversely holomorphic foliation on a compact complex manifold, exhibits a compact stable leaf if and only if the set of compact leaves is not a zero measure subset of the manifold.
Deformation of morphisms along leaves of foliations define the tangential foliation on the corresponding space of morphisms. We prove that codimension one fo-liations having a tangential foliation with at least one non-algebraic leaf are…
In this paper, we study transversely holomorphic flows, i.e. those whose holonomy pseudo-group is given by biholomorphic maps. We prove that for Anosov flows on smooth compact manifolds, the strong unstable (respectively, stable)…
We prove that any holomorphic locally homogeneous geometric structure on a complex torus, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true is any dimension. In higher dimension we…
We give sufficient conditions for the tautness of a transversely homogenous foliation defined on a compact manifold, by computing its base-like cohomology. As an application, we prove that if the foliation is non-unimodular then either the…
We construct integrable holomorphic G-structures and flat holomorphic Cartan geometries on every complex Hopf manifold, without using the normal forms given by the Poincar\'e-Dulac Theorem. We provide a new proof of the latter using charts…
On every compact and orientable three-manifold, we construct total foliations (three codimension 1 foliations that are transverse at every point). This construction can be performed on any homotopy class of plane fields with vanishing Euler…
We classify the holomorphic parabolic geometries on compact complex manifolds of general type. We accomplish this by bounding the numerical dimension of any smooth projective variety in terms of geometric invariants of the flag variety…