Complex submanifolds, connections and asymptotics

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.

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.

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 $$k\in {\mathbb {N}}$$ .

We prove a classification theorem by cohomology classes for compact Riemannian manifolds with a one-parameter group of isometries without fixed points generalizing the classification of line bundles (more precisely, their circle bundles) over compact manifolds by their first Chern class. We also prove a classification theorem generalizing that of holomorphic line bundles over compact complex manifold by the Picard group of the base for a subfamily of manifolds with additional structure resembling that of circle bundles of such holomorphic line bundles.

A note on the Bergman Kernel

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

Let L L be a line bundle over a compact complex manifold X X and denote by h L h_{L} and h X h_{X} fixed Hermitian metrics on L L and T X , TX, respectively. Our main result provides a formula for the average distribution of the exponentially small eigenvalues of the corresponding Dolbeault Laplacians associated to high tensor powers of L , L, which in physics terminology is a measure of “tunneling” of the Dolbeault complex. Along the way a new proof of the asymptotics of the induced Quillen metric on the corresponding determinant line is obtained. A brief comparison with the tunneling effect for Witten Laplacians and large deviation principles for fermions is also made.

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.

Let L ¯ i →X i be a holomorphic line bundle over a compact complex manifold for i=1,2. Let S i denote the associated principal circle-bundle with respect to some hermitian inner product on L ¯ i . We construct complex structures on S=S 1 ×S 2 which we refer to as scalar, diagonal, and linear types. While scalar type structures always exist, the more general diagonal but non-scalar type structures are constructed assuming that L ¯ i are equivariant (ℂ * ) n i -bundles satisfying some additional conditions. The linear type complex structures are constructed assuming X i are (generalized) flag varieties and L ¯ i negative ample line bundles over X i . When H 1 (X 1 ;ℝ)=0 and c 1 (L ¯ 1 )∈H 2 (X 1 ;ℂ) is non-zero, the compact manifold S does not admit any symplectic structure and hence it is non-Kähler with respect to any complex structure.We obtain a vanishing theorem for H q (S;𝒪 S ) when X i are projective manifolds, L ¯ i ∨ are very ample and the cone over X i with respect to the projective imbedding defined by L ¯ i ∨ are Cohen-Macaulay. We obtain applications to the Picard group of S. When X i =G i /P i where P i are maximal parabolic subgroups and S is endowed with linear type complex structure with “vanishing unipotent part” we show that the field of meromorphic functions on S is purely transcendental over ℂ.

Many problems in differential and analytic geometry seem to have something to do with the study of non-linear partial differential equations of elliptic type. For instance, the classical Weyl and Minkowski problems for a convex surface have been studied by H. Weyl, H. Lewy, and L. Nirenberg using the iteration method for the construction of solutions of certain non-linear equations of elliptic type (see [6]). Also, M. Kuranishi [3] constructed the effective complete family of deformations of complex analytic structures on a given compact complex manifold as the solution space of another non-linear equation of elliptic type; whereby the basic idea in his work is to apply an implicit function theorem to the non-linear operator of a Banach space, and to construct the bifurcation of solutions explicitly.

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.

Let $\pi \colon X \to B$ be a holomorphic submersion between compact K\ahler manifolds of any dimensions, whose fibres and base have no non-zero holomorphic vector fields and whose fibres admit constant scalar curvature K\ahler metrics. This article gives a sufficient topological condition for the existence of a constant scalar curvature K\ahler metric on $X$. The condition involves the $\CM$-line bundle---a certain natural line bundle on $B$---which is proved to be nef. Knowing this, the condition is then implied by $c_1(B) <0$. This provides infinitely many K\ahler manifolds of constant scalar curvature in every dimension, each with K\ahler class arbitrarily far from the canonical class.

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.