Holomorphic Manifolds Research Articles

Holomorphic manifolds on locally convex spaces

Based on locally compact perturbations of the identity map similar to the Fredholm structures on real Banach manifolds, complex manifolds with inverse mapping theorem as part of the defintion are proposed. Standard topics including holomorphic maps, morphisms, derivatives, tangent bundles, product manifolds and submanifolds are presented. Although this framework is elementary, it lays the necessary foundation for all subsequent developments.

Classes de Chern des ensembles analytiques

Let V be a compact complex analytical subset of a holomorphic manifold M. We shall define classes in homology, which coincide, when V is non-singular, with the Poincaré duals c ∗(N V)⌢[V] and c ∗(TV)⌢[V] of the Chern classes of the normal bundle N V and of the tangent bundle TV. However, these definitions depend in general on the data on a desingularization ϕ :V′→V of V, except in some particular cases, as complex curves or sets which are locally complete intersection (LCI). These classes make possible to generalize some theories already known for LCI, such as the various indices of foliations relatively to invariant subsets, or the Minor numbers and classes. To cite this article: V. Cavalier et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Poletsky theory of disks on holomorphic manifolds

Poletsky's construction of plurisubharmonic functions by a minimum principle is generalized to arbitrary complex manifolds.

The manifold of finite rank projections in the algebra ℒ(H) of bounded linear operators

Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V = ℒ( H). Using the algebraic structure of V, a torsionfree affine connection ∇ (that is invariant under the group of automorphisms of V) is defined on every connected component m of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0 ≤ r ≤ ∞). We prove that m admits a Riemann structure if and only if m consists of projections that have the same finite rank r or the same finite corank, and in that case ∇ is the Levi-Civita and the Kähler connection of m. Moreover, m turns out to be a totally geodesic Riemann manifold whose geodesics and Riemann distance are computed.

Projective spaces of aC *-algebra

Based on the projective matrix spaces studied by B. Schwarz and A. Zaks, we study the notion of projective space associated to a C*-algebraA with a fixed projectionp. The resulting spaceP(p) admits a rich geometrical structure as a holomorphic manifold and a homogeneous reductive space of the invertible group ofA. Moreover, several metrics (chordal, spherical, pseudo-chordal, non-Euclidean-in Schwarz-Zaks terminology) are considered, allowing a comparison amongP(p), the Grassmann manifold ofA and the space of positive elements which are unitary with respect to the bilinear form induced by the reflection ε=2p−1. Among several metrical results, we prove that geodesics are unique and of minimal length when measured with the spherical and non-Euclidean metrics.

On ${\mathbf{R}}^{+}$ and ${\mathbf{C}}$ complete holomorphic vector fields

We show that, on holomorphic manifolds that have a plurisubharmonic exhaustion function and that do not carry nonconstant bounded plurisubharmonic functions (e.g. Ca), holomorphic vector fields that are complete in positive time are complete in complex time.

Remarks on the generic rank of a CR mapping

We study germs of smooth CR mappings between embedded real hypersurfaces in complex spaces of the same dimension. In particular, we are interested in the generic rank of such mappings. IfH:M →M′ is a CR map between two hypersurfacesM andM′, we prove that ifM′ does not contain any germ of a holomorphic manifold then eitherH is constant or the generic rank ofH is odd. We also prove that if there is no formal holomorphic vector field tangent toM, then eitherH is constant or genericallyH is a local diffeomorphism. It follows, as a special case, that ifM andM′ are of D-finite type (in the sense of D’Angelo) thenH is either constant or is generically a local diffeomorphism.