Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds

It is well known that a pseudo-Kähler structure is one of the natural generalizations of a Kähler structure. In this paper, we consider the Dolbeault cohomology groups of compact pseudo-Kähler homogeneous manifolds.     

Ricci flatness of certain compact pseudo-Kähler solvmanifolds

It is well known that a pseudo-Kähler structure is one of the natural generalizations of a Kähler structure. In this paper, we consider Ricci curvature tensor of certain compact pseudo-Kähler solvmanifolds.     

An invariant formula for a star product with separation of variables

We give an invariant formula for a star product with separation of variables on a pseudo-Kähler manifold.     

Warped product Kähler manifolds and Bochner–Kähler metrics

Using as an underlying manifold an alpha-Sasakian manifold, we introduce warped product Kähler manifolds. We prove that if the underlying manifold is an alpha-Sasakian space form, then the corresponding Kähler manifold is of quasi-constant holomorphic sectional curvatures with a special distribution. Conversely, we prove that any Kähler manifold of quasi-constant holomorphic sectional curvatures with a special distribution locally has the structure of a warped product Kähler manifold whose base is an alpha-Sasakian space form. As an application, we describe explicitly all Bochner–Kähler metrics of quasi-constant holomorphic sectional curvatures. We find four families of complete metrics of this type. As a consequence, we obtain Bochner–Kähler metrics generated by a potential function of distance in complex Euclidean space and of time-like distance in the flat Kähler–Lorentz space.     

Kähler metrics generated by functions of the time-like distance in the flat Kähler–Lorentz space

We prove that every Kähler metric, whose potential is a function of the time-like distance in the flat Kähler–Lorentz space, is of quasi-constant holomorphic sectional curvatures, satisfying certain conditions. This gives a local classification of the Kähler manifolds with the above-mentioned metrics. New examples of Sasakian space forms are obtained as real hypersurfaces of a Kähler space form with special invariant distribution. We introduce three types of even dimensional rotational hypersurfaces in flat spaces and endow them with locally conformal Kähler structures. We prove that these rotational hypersurfaces carry Kähler metrics of quasi-constant holomorphic sectional curvatures satisfying some conditions, corresponding to the type of the hypersurfaces. The meridians of those rotational hypersurfaces, whose Kähler metrics are Bochner–Kähler (especially of constant holomorphic sectional curvatures), are also described.     

On holomorphic maps and Generalized Complex Geometry

We introduce a natural notion of holomorphic map between generalized complex manifolds and we prove some related results on Dirac structures and generalized Kähler manifolds.     

HyperKähler and quaternionic Kähler manifolds with S -symmetries

We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKähler manifold admitting hyperKähler potential and triholomorphic action of S¹$S^1$ can be constructed from another hyperKähler manifold (of lower dimension) with an action of S¹$S^1$ that fixes one complex structure and rotates the other two and vice versa. We also study the corresponding quaternionic Kähler manifolds equipped with a quaternionic Kähler action of the circle. In particular we show that any positive quaternionic Kähler manifolds with S¹$S^1$-symmetry admits a Kähler metric on an open everywhere dense subset.     

Deformation quantization of nonholonomic almost Kähler models and Einstein gravity

Nonholonomic distributions and adapted frame structures on (pseudo) Riemannian manifolds of even dimension are employed to build structures equivalent to almost Kähler geometry and which allows to perform a Fedosov-like quantization of gravity. The nonlinear connection formalism that was formally elaborated for Lagrange and Finsler geometry is implemented in classical and quantum Einstein gravity.     

Homogeneous Kähler manifolds: Paving the way towards new supersymmetric sigma models

Homogeneous Kähler manifolds give rise to a broad class of supersymmetric sigma models containing, as a rather special subclass, the more familiar supersymmetric sigma models based on Hermitian symmetric spaces. In this article, all homogeneous Kähler manifolds with semisimple symmetry groupG are constructed, and are classified in terms of Dynkin diagrams. Explicit expressions for the complex structure and the Kähler structure are given in terms of the Lie algebra g ofG. It is shown that for compactG, one can always find an Einstein-Kähler structure, which is unique up to a constant multiple and for which the Kähler potential takes a simple form.On leave of absence from Fakultät für Physik der Universität Freiburg, FRG     

Ruled CR-submanifolds of locally conformal Kähler manifolds

The purpose of this paper is to study the canonical totally real foliations of CR-submanifolds in a locally conformal Kähler manifold.     

Hyper-Kähler induction and self-duality on Riemann surfaces

Universal hyper-Kähler spaces are constructed from Lie groups acting on flat Kähler manifolds. These spaces are used to describe the moduli space of solutions of Hitchin's equation — self-duality equations on a Riemann surface — as the contangent bundle of the moduli space of flat connections on a Riemann surface.     

Hamiltonian constructions of Kähler-Einstein metrics and Kähler metrics of constant scalar curvature

Assuming the existence of a real torus acting through holomorphic isometries on a Kähler manifold, we construct an ansatz for Kähler-Einstein metrics and an ansatz for Kähler metrics with constant scalar curvature. Using this Hamiltonian approach we solve the differential equations in special cases and find, in particular, a family of constant scalar curvature Kähler metrics describing a non-linear superposition of the Bergman metric, the Calabi metric and a higher dimensional generalization of the LeBrun Kähler metric. The superposition contains Kähler-Einstein metrics and all the geometries are complete on the open disk bundle of some line bundle over the complex projective spaceP _n. We also build such Kähler geometries on Kähler quotients of higher cohomogeneity.Partially supported by the NSF Under Grant No. DMS 8906809     

Coherent states in geometric quantization

In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kähler and non-Kähler cases. We then study the reproducing kernels associated to the quantum Hilbert spaces and use them to define symplectic coherent states. The rest of the paper is dedicated to the properties of symplectic coherent states and the corresponding Berezin–Toeplitz quantization. Specifically, we study overcompleteness, symplectic analogues of the basic properties of Bargmann’s weighted analytic function spaces, and the ‘maximally classical’ behavior of symplectic coherent states. We also find explicit formulas for symplectic coherent states on compact Riemann surfaces.     

First Eigenvalue of the Dirac Operator for a Kähler Manifold of Even Complex Dimension: The Limiting Case

K. D. Kirchberg has given a minoration of the absolute value of the eigenvalues of the Dirac operator for a compact Kähler spin manifold (W,g) with positive scalar curvature and has introduced, in this context, the notion of Kähler twistor-spinor. We prove here that if dim_C W = p 4 is even, in the limiting case, (W, g) is the Kähler product of an odd-dimensional limiting case compact Kähler spin manifold of complex dimension (p-1), by a flat Kähler manifold which is a compact quotient of C.     

On complex Landsberg and Berwald spaces

In this paper, we study complex Landsberg spaces and some of their important subclasses. The tools of this study are the Chern-Finsler, Berwald, and Rund complex linear connections. We introduce and characterize the class of generalized Berwald and complex Landsberg spaces. The intersection of these spaces gives the so-called G-Landsberg class. This last class contains two other kinds of complex Finsler spaces: strong Landsberg and G-Kähler spaces. We prove that the class of G-Kähler spaces coincides with complex Berwald spaces, in Aikou’s (1996) [1] sense, and it is a subclass of the strong Landsberg spaces. Some special complex Finsler spaces with (α,β)-metrics offer examples of generalized Berwald spaces. Complex Randers spaces with generalized Berwald and weakly Kähler properties are complex Berwald spaces.     

Clifford algebra and the propagation of Kähler spinors

The Kähler equation for an inhomogeneous differential form is analyzed in some detail and expressed in a set of coordinates called Riemann normal coordinates. A class of solutions to the Kähler spinors is constructed. It is shown how we can perturbatively decouple the Kähler equation and write its solution as a sum of spinors by considering the isomorphism between Clifford and the total matrix algebras.     

Einstein gravity in almost Kähler and Lagrange–Finsler variables and deformation quantization

A geometric procedure is elaborated for transforming (pseudo) Riemannian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented as almost Kähler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2$2+2$ splitting with associate nonlinear connection structure. We also show how the Einstein equations can be written in terms of Lagrange–Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.     

Random Kähler metrics

The purpose of this article is to propose a new method to define and calculate path integrals over metrics on a Kähler manifold. The main idea is to use finite dimensional spaces of Bergman metrics, as an approximation to the full space of Kähler metrics. We use the theory of large deviations to decide when a sequence of probability measures on the spaces of Bergman metrics tends to a limit measure on the space of all Kähler metrics. Several examples are considered.