Sorin Dumitrescu

Structures géométriques holomorphes sur les variétés complexes compactes

Abstract We study holomorphic geometric structures on compact complex manifolds. We show that, contrary to the situation in the real domain, a holomorphic geometric structure on a compact complex manifold usually admits a “big” pseudogroup of local isometries. We exhibit very general conditions which imply that the pseudogroup of local isometries acts transitively.

QUASIHOMOGENEOUS ANALYTIC AFFINE CONNECTIONS ON SURFACES

We classify torsion-free real-analytic affine connections on compact oriented real-analytic surfaces which are locally homogeneous on a nontrivial open set, without being locally homogeneous on all of the surface. In particular, we prove that such connections exist. This classification relies on a local result that classifies germs of torsion-free real-analytic affine connections on a neighborhood of the origin in the plane which are quasihomogeneous, in the sense that they are locally homogeneous on an open set containing the origin in its closure, but not locally homogeneous in the neighborhood of the origin.

Quasihomogeneous three-dimensional real-analytic Lorentz metrics do not exist

We show that a germ of a real-analytic Lorentz metric on

Symmetries of holomorphic geometric structures on tori

{\mathbb R}^3

Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds

R3 which is locally homogeneous on an open set containing the origin in its closure is necessarily locally homogeneous. We classifiy Lie algebras that can act quasihomogeneously—meaning they act transitively on an open set admitting the origin in its closure, but not at the origin—and isometrically for such a metric. In the case that the isotropy at the origin of a quasihomogeneous action is semisimple, we provide a complete set of normal forms of the metric and the action.

Géométries Lorentziennes de dimension 3 : classification et complétude

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 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).

Homogénéité locale pour les métriques riemanniennes holomorphes en dimension

Abstract We prove that any holomorphic locally homogeneous geometric structure on a complex torus of dimension two, modelled on a complex homogeneous surface, is translation invariant. We conjecture that this result is true in any dimension. In higher dimension, we prove it for G nilpotent. We also prove that for any given complex algebraic homogeneous space (X, G), the translation invariant (X, G)-structures on tori form a union of connected components in the deformation space of (X, G)-structures.

3

This is a survey paper dealing with quasihomogeneous geometric structures, in the sense that they are locally homogeneous on a nontrivial open set, but not on all of the manifold. Our motivation comes from Gromov’s open-dense orbit theorem which asserts that if the pseudogroup of local automorphisms of a rigid geometric structure acts with a dense orbit, then this orbit is open. Fisher conjectured that the maximal open set of local homogeneity is all of the manifold as soon as the following three conditions are fulfilled: the automorphism group of the manifold acts with a dense orbit, the geometric structure is a G-structure (meaning that it is locally homogeneous at the first order) and the manifold is compact. In a recent joint work, with Adolfo Guillot, we succeeded to prove Fisher’s conjecture for real analytic torsion free affine connections on surfaces: we construct and classify those connections which are quasihomogeneous; their automorphism group never act with a dense orbit.