The Levi problem on strongly pseudoconvex G-bundles

Let G be a unimodular Lie group, X a compact manifold with boundary, and M the total space of a principal bundle G → M → X so that M is also a strongly pseudoconvex complex manifold. In this study, we show that if G acts by holomorphic transformations satisfying a local property, then the space of square-integrable holomorphic functions on M is infinite-dimensional.     

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

We study compact complex 3-manifolds M admitting a (locally homogeneous) holomorphic Riemannian metric g. We prove the following: (i) If the Killing Lie algebra of g has a non trivial semi-simple part, then it preserves some holomorphic Riemannian metric on M with constant sectional curvature; (ii) If the Killing Lie algebra of g is solvable, then, up to a finite unramified cover, M is a quotient Γ\G, where Γ is a lattice in G and G is either the complex Heisenberg group, or the complex SOL group. S. Dumitrescu was partially supported by the ANR Grant BLAN 06-3-137237.     

Holomorphic Functions of Exponential Growth on Abelian Coverings of a Projective Manifold

Let M be a projective manifold, p: M _G M a regular covering over M with a free Abelian transformation group G. We describe the holomorphic functions on M _G of an exponential growth with respect to the distance defined by a metric pulled back from M. As a corollary, we obtain Cartwright and Liouville-type theorems for such functions. Our approach brings together the L ₂ cohomology technique for holomorphic vector bundles on complete Kähler manifolds and the geometric properties of projective manifolds.     

Nonnegatively curved fixed point homogeneous 5-manifolds

Let G be a compact Lie group acting effectively by isometries on a compact Riemannian manifold M with nonempty fixed point set Fix(M, G). We say that the action is fixed point homogeneous if G acts transitively on a normal sphere to some component of Fix(M, G), equivalently, if Fix(M, G) has codimension one in the orbit space of the action. We classify up to diffeomorphism closed, simply connected 5-manifolds with nonnegative sectional curvature and an effective fixed point homogeneous isometric action of a compact Lie group.     

Version kählérienne d'une conjecture de Robert J. Zimmer

Let M be a compact Kähler manifold. Let G be a connected simple real Lie group. Let Γ be a lattice in G. We prove the following: if the R-rank of G is strictly larger than the complex dimension of M any morphism from Γ to the group of holomorphic diffeomorphisms of M has finite image. This is a particular case in a conjecture of Robert J. Zimmer     

Holomorphic Differential Operators and Plane Sets with Dense Images

We prove in this paper that, given a nonempty open set G in the complex plane, a subset A of G which is not relatively compact and a holomorphic infinite order differential or antidiffeärential operator T, then there are holomorphic functions ? on G such that the image of A under T ? is dense in the complex plane. This extends a recent result about a property of boundary behaviour exhibited by the derivative operator.     

Simplicity of stable principal sheaves

Let M be a compact connected Kähler manifold, and let Gbe a connected complex reductive linear algebraic group. Weprove that a principal G-sheaf on M admits an admissible Einstein–Hermitianconnection if and only if the principal G-sheaf is polystable.Using this it is shown that the holomorphic sections of theadjoint vector bundle of a stable principal G-sheaf on M aregiven by the center of the Lie algebra of G. The Bogomolov inequalityis shown to be valid for polystable principal G-sheaves.     

Holomorphic Cartan geometry on manifolds with numerically effective tangent bundle

Let X be a compact connected Kähler manifold such that the holomorphic tangent bundle TX is numerically effective. A theorem of Demailly et al. (1994) [11] says that there is a finite unramified Galois covering M→X, a complex torus T, and a holomorphic surjective submersion f:M→T, such that the fibers of f are Fano manifolds with numerically effective tangent bundle. A conjecture of Campana and Peternell says that the fibers of f are rational and homogeneous. Assume that X admits a holomorphic Cartan geometry. We prove that the fibers of f are rational homogeneous varieties. We also prove that the holomorphic principal G-bundle over T given by f, where G is the group of all holomorphic automorphisms of a fiber, admits a flat holomorphic connection.     

The Chern character of a transversally elliptic symbol and the equivariant index

Let G be a compact Lie group acting on a compact manifold M. In this article, we associate to a G-transversally elliptic symbol on M a G-invariant generalized function on G, constructed in terms of equivariant closed differential forms on the cotangent bundle T ^* M. Oblatum 24-VII-1995     

Classical theorems of probability on Gelfand pairs—Khinchin’s theorems and Cramér’s theorem

We prove Khinchin’s Theorems for Gelfand pairs (G, K) satisfying a condition (*): (a)G is connected; (b)G is almost connected and Ad (G/M) is almost algebraic for some compact normal subgroupM; (c)G admits a compact open normal subgroup; (d) (G,K) is symmetric andG is 2-root compact; (e)G is a Zariski-connectedp-adic algebraic group; (f) compact extension of unipotent algebraic groups; (g) compact extension of connected nilpotent groups. In fact, condition (*) turns out to be necessary and sufficient forK-biinvariant measures on aforementioned Gelfand pairs to be Hungarian. We also prove that Cramér’s theorem does not hold for a class of Gaussians on compact Gelfand pairs. This author was supported by the European Commission (TMR 1998–2001 Network Harmonic Analysis).     

The Prym and Ganning Vector Bundles over the Teichmueller Space

We introduce and study the Prym vector bundle P of holomorphic Prym differentials and the Ganning cohomology bundle G over the Teichmueller space of compact Riemann surfaces of genus g2 and over the Torelli space of genus g2. We construct a basis of holomorphic Prym differentials on a variable compact Riemann surface which depends on the moduli of the compact Riemann surface and on the essential characters. From these bundles we compose an exact sequence of holomorphic vector bundles over the product of the Teichmueller space of genus g and a special domain in the complex manifold C ^2g/Z ^2g.     

Holomorphic maps of Kähler manifolds with constrained energy density

We consider holomorphic and antiholomorphic maps of Kähler manifoldsM andN withM compact. In view of bounds on the Ricci curvature ofM and the holomorphic bisectional curvature ofN, the energy density of the map is constrained to satisfy certain inequalities. One inequality implies that the map is constant. Another specifies the image ofM as a totally geodesic real surface of constant Gaussian curvature inN.     

Homogeneous Riemannian manifolds of positive Ricci curvature

We prove that a homogeneous effective spaceM=G/H, whereG is a connected Lie group andH⊂G is a compact subgroup, admits aG-invariant Riemannian metric of positive Ricci curvature if and only if the spaceM is compact and its fundamental group π₁(M) is finite (in this case any normal metric onG/H is suitable). This is equivalent to the following conditions: the groupG is compact and the largest semisimple subgroupLG⊂G is transitive onG/H. Furthermore, ifG is nonsemisimple, then there exists aG-invariant fibration ofM over an effective homogeneous space of a compact semisimple Lie group with the torus as the fiber. Translated fromMatematicheskie Zametki, Vol. 58, No. 3, pp. 334–340, September, 1995.     

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Published online: 1 February 2002     

Equivariant holomorphic Hermitian principal bundles over a generalized flag manifold

Let G be a simple and simply connected complex linear algebraic group. Fix a maximal compact subgroup K(G) ì G{K(G) \subset G}, and let P be a parabolic subgroup of G. Let H be any connected reductive complex linear algebraic group. We classify the K(G)-equivariant holomorphic Hermitian principal H-bundles over G/P.     

Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S¹ ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.     

Hypercomplex structures on Kähler manifolds

Let (M, I) be a compact K?hler manifold admitting a hypercomplex structure (M, I, J, K). We show that (M, I, J, K) admits a natural HKT-metric. This is used to construct a holomorphic symplectic form on (M, I). Received: August 2004 Revision: August 2004 Accepted: August 2004     

Amenability of the Second Dual of Hypergroup Algebras

The amenability of the Banach algebra L ¹(G), the measure algebra M(G) and their second duals of a locally compact group have been considered by a number of authors. During these investigations it has been shown that L ¹(G)^** is amenable if and only if G is finite. If LUC (G)^*, the dual of the space of left uniformly continuous functions on G, is amenable, then G is compact and M(G) is amenable. Finally, if M(G)^** is amenable, then G is finite. The aim of this paper is to generalize all of the above results to the locally compact hypergroups.     

Dolbeault cohomology of compact nilmanifolds

LetM=G/ be a compact nilmanifold endowed with an invariant complex structure. We prove that on an open set of any connected component of the moduli space of invariant complex structures onM, the Dolbeault cohomology ofM is isomorphic to the cohomology of the differential bigraded algebra associated to the complexification of the Lie algebra ofG. to obtain this result, we first prove the above isomorphism for compact nilmanifolds endowed with a rational invariant complex structure. This is done using a descending series associated to the complex structure and the Borel spectral sequences for the corresponding set of holomorphic fibrations. Then we apply the theory of Kodaira-Spencer for deformations of complex structures.Research partially supported by MURST and CNR of Italy.Research partially supported by MURST and CNR of Italy.