We prove that any compact Kähler manifold bearing a holomorphic Cartan geometry contains a rational curve just when the Cartan geometry is inherited from a holomorphic Cartan geometry on a lower dimensional compact Kähler manifold. This shows that many complex manifolds admit no or few holomorphic Cartan geometries.
The aim of this paper is to present examples of holomorphically pseudosymmetric Kähler metrics on the complex projective spaces ℂℙⁿ, where n ≥ 2.
In this note we prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kähler metric on a compact Hermitian symmetric spaces of ABCD–type.
Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant...
Dans cet article nous nous intéressons aux immersions isométriques minimales (resp. pluriharmoniques) définies sur une variété riemannienne munie d’une 2-forme parallèle non triviale à valeurs dans une variété riemannienne ou kählérienne de courbure isotrope négative (resp. positive). Les résultats que nous obtenons généralisent certains résultats bien connus de non existence et de rigidité concernant les immersions minimales et pluriharmoniques de variétés kählériennes dans les espaces formes réels...
In this note it is shown that almost Hermitian locally homogeneous manifolds are determined, up to local isometries, by an integer , the covariant derivatives of the curvature tensor up to order and the covariant derivatives of the complex structure up to the second order calculated at some point. An example of a Hermitian locally homogeneous manifold which is not locally isometric to any Hermitian globally homogeneous manifold is given.
This paper develops various estimates for solutions of a nonlinear, fouth order PDE which corresponds to prescribing the scalar curvature of a toric Kähler metric. The results combine techniques from Riemannian geometry and from the theory of Monge-Ampère equations.