A note on the Bergman Kernel

We study the (k, s)-positivity for holomorphic vector bundles on compact complex manifolds. (0, s)-positivity is exactly the Demailly s-positivity and a (k, 1)-positive line bundle is just a k-positive line bundle in the sense of Sommese. In this way we get a unified theory for all kinds of positivities used for semipositive vector bundles. Several new vanishing theorems for (k, s)-positive vector bundles are proved and the vanishing theorems for k-ample vector bundles on projective algebraic manifolds are generalized to k-positive vector bundles on compact Kähler manifolds.

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers M N of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns “random” sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {S N j} of H 0(M, L N ), we show that for almost every sequence {S N j}, the associated sequence of zero currents &1/N Z S N j ; tends to the curvature form ω of L. Thus, the zeros of a sequence of sections s N ∈H 0(M, L N ) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {S N j } of H 0(M, L N ) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.

Let L → X be a positive line bundle on a compact complex manifold X. For compact submanifolds Y, S of X and a holomorphic submersion Y → S with compact fibre, we study curvature of a natural connection on certain line bundles on S.

We describe the norming sets for the space of global holomorphic sections to a k-power of a positive holomorphic line bundle on a compact complex manifold X. We characterize in metric terms the sequence of measurable subsets $$\{G_{k}\}_{k}$$ of X such that there is a constant $$C > 0$$ where $$\begin{aligned} \Vert s\Vert ^{2}\le C \int _{G_{k}} |s(z)|^{2} \mathrm{{d}}V(z) \end{aligned}$$ for every $$s\in H^{0}(X,{\mathcal {O}}(L^{\otimes k}))$$ and for all $$k\in {\mathbb {N}}$$ .

Let (E, h) be a semipositive hermitian holomorphic line bundle on a compact complex manifold X with dimX>11$$\\end{document}]]>. Assume that for each point x∈X there exists a divisor D∈|E| in the complete linear system determined by E whose complement X\\D is a Stein neighbourhood of x with the density property. Then, the disc bundle Δh(E)={e∈E:|e|h<1} is an Oka manifold while Dh(E)={e∈E:|e|h>1}1\\}$$\\end{document}]]> is a Kobayashi hyperbolic domain. In particular, the zero section of E admits a basis of Oka neighbourhoods {|e|h<c} with c>00$$\\end{document}]]>. We show that this holds if X is a rational homogeneous manifold of dimension >11$$\\end{document}]]>. This class of manifolds includes complex projective spaces, Grassmannians, and flag manifolds. This phenomenon contributes to the heuristic principle that Oka properties are related to metric positivity of complex manifolds.

Scalar curvature and holomorphy potentials

We study two natural problems concerning the scalar and the Ricci curvatures of the Bismut connection. Firstly, we study an analog of the Yamabe problem for Hermitian manifolds related to the Bismut scalar curvature, proving that, fixed a conformal Hermitian structure on a compact complex manifold, there exists a metric with constant Bismut scalar curvature in that class when the expected constant scalar curvature is non-negative. A similar result is given in the general case of Gauduchon connections. We then study an Einstein-type condition for the Bismut Ricci curvature tensor on principal bundles over Hermitian manifolds with complex tori as fibers. Thanks to this analysis, we construct explicit examples of Calabi–Yau with torsion Hermitian structures and prove a uniqueness result for them.

Transposing the Berezin quantization into the setting of analytic microlocal analysis, we construct approximate semiclassical Bergman projections on weighted $L^{2}$ -spaces with analytic weights, and show that their kernel functions admit an asymptotic expansion in the class of analytic symbols. As a corollary, we obtain new estimates for asymptotic expansions of the Bergman kernel on $\\mathbb{C}^{n}$ and for high powers of ample holomorphic line bundles over compact complex manifolds.

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich–Wasserstein distance of the Fekete points to the limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.

Bergman kernels related to Hermitian line bundles over compact complex manifolds by B. Berndtsson A gentle introduction to points of finite type on real hypersurfaces by J. P. D'Angelo The moduli space of 2-step nilpotent Lie algebras of type $(p,q)$ by P. Eberlein The homotopy principle in complex analysis: A survey by F. Forstneric Finiteness theorems in riemannian geometry by K. Grove Generalized Toponogov's theorem for manifolds with radial curvature bounded below by Y. Itokawa, Y. Machigashira, and K. Shiohama The global isometric embedding problem by H. Jacobowitz The Bergman metric invariants and their boundary behavior by K.-T. Kim and S. G. Krantz Natural connections in almost complex manifolds by S. Kobayashi The generalized triangle inequalities for rank 3 symmetric spaces of noncompact type by S. Kumar, B. Leeb, and J. Millson Subelliptic estimates and scaling in the $\bar\partial$-Neumann problem by J. D. McNeal Negativity of curvature on spaces parametrizing Hodge decompositions of reduced first cohomology groups by N. Mok On the extension of $L^2$ holomorphic functions VI--A limiting case by T. Ohsawa Variations on a theme of Synge by P. Petersen Mappings between real submanifolds in complex space by L. P. Rothschild Harmonic analysis on toric varieties by B. Shiffman, T. Tate, and S. Zelditch On complex manifolds with noncompact automorphism groups by B. Wong Kahler manifolds with slightly positive bisectional curvature by H.-H. Wu and F. Zheng.

This is a survey dealing with holomorphic G-structures and holomorphic Cartan geometries on compact complex manifolds. Our emphasis is on the foliated case. We investigate holomorphic foliations with a transverse holomorphic Cartan geometry, and also with the more general structure of branched transverse holomorphic Cartan geometry.The first part of the chapter presents the geometric notion of holomorphic G-structure whose origin was motivated by various classical examples. We explain some classification results for compact complex manifolds endowed with holomorphic G-structures highlighting the special case of \(\mathrm {GL}(2,{\mathbb {C}})\)-structures and \(\mathrm {SL}(2,{\mathbb {C}})\)-structures. The second part of the survey deals with holomorphic Cartan geometry in the classical case and in the branched and generalized case. Two definitions of foliated (branched, generalized) Cartan geometry are described and shown to be equivalent. We provide some classification results of compact complex manifolds with foliated (branched or generalized) Cartan geometries. At the end some related open problems are formulated.KeywordsHolomorphic geometric structureHolomorphic GL(2)-structureKähler manifoldFoliated principal bundleFoliated Cartan geometry1991 Mathematics Subject Classification53C0753C1032Q57

The aim of this paper is to collect some interesting unsolved problems in the topology of compact complex manifolds which are scattered in literature; needless to remark, some of them are known to specialists. Besides, the readermay consult the well-known list of problems by F. Hirzebruch [ 5 ] which has stimulated remarkable progress in this direction. But the topology of complex manifolds has always baffled workers in this field and is still far from satisfactory. Of course, there is remarkable progress in the case of homogeneous compact complex manifolds. We start with the following result.

In the first half of the twentieth century, new tools were developed for the study of the algebraic topology of manifolds, and in particular, complex manifolds: de Rham’s representation of algebraic topology with differential forms, Kahler’s introduction of specific types of Riemannian metrics for complex manifolds, and Hodge’s theory of harmonic differential forms. These tools, along with the theory of sheaf cohomology developed by Leray somewhat later, allowed Kodaira to formulate and prove in 1954 an embedding theorem for compact complex manifolds which completely characterized the abstract manifolds that could be embedded as a submanifold of a complex projective space.

We consider various types of special Hermitian metrics on compact complex manifolds, namely Kahler, Hermitian-symplectic, SKT, balanced and strongly Gauduchon metrics. We study the strongly Gauduchon metrics in relation to the SKT condition on complex nilmanifolds and to the Hermitian-symplectic condition on certain complex solvmanifolds. We also review the behaviour of the existence properties of these special metrics on compact complex manifolds under holomorphic deformations.

We find the precise rate at which the empirical measure associated to a β-ensemble converges to its limiting measure. In our setting the β-ensemble is a random point process on a compact complex manifold distributed according to the β power of a determinant of sections in a positive line bundle. A particular case is the spherical ensemble of generalized random eigenvalues of pairs of matrices with independent identically distributed Gaussian entries.