arXiv research

In this paper, by using analytical methods we obtain a generalization of the famous Kodaira embedding theorem.

In this paper, we survey some recent results about the asymptotic expansion of Bergman kernel and we give a Bergman kernel proof of Kodaira embedding theorem.

Explains Hodge theory and Kodaira embedding theorem for complex manifolds.

We consider a compact connected CR manifold with a transversal CR locally free $\mathbb R$-action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion and we establish $\mathbb R$-equivariant K…

We define analogue of theta-functions on the Kodaira--Thurston manifold which is a compact 4-dimensional symplectic manifold and use them to construct canonical symplectic embedding of the Kodaira--Thurston manifold into the complex projective space (analogue of the Lefshetz theorem).

We consider a compact CR manifold with a transversal CR locally free circle action endowed with a rigid positive CR line bundle. We prove that a certain weighted Fourier-Szegő kernel of the CR sections in the high tensor powers admits a full asymptotic expansion. As a consequence, we establish an equivariant Kodaira em…

We give several generalizations of the Kodaira vanishing and embedding theorems for Kähler manifolds to the case where the relevent line bundle has a small region of negative curvature. To prove the vanishing theorems we adapt techniques of Elworthy-Rosenberg for vanishing theorems in Riemannian geometry. For the embed…

Let $X$ be an orientable compact Levi-flat CR manifold and let $L$ be a positive CR complex line bundle over $X$. We prove that certain microlocal conjugations of the associated Szegő kernel admits an asymptotic expansion with respect to high powers of $L$. As an application, we give a Szegő kernel proof of the Kodaira…

Study Kodaira dimensions on almost complex manifolds, proving integrability and structural descriptions.

Kodaira embedding theorem provides an effective characterization of projectivity of a Kähler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kähler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms…

This paper proves Hölder continuity for complex Monge-Ampère equations on Kähler varieties.

Let $X$ be an abstract not necessarily compact orientable CR manifold of dimension $2n-1$, $n\geqslant2$, and let $L^k$ be the $k$-th tensor power of a CR complex line bundle $L$ over $X$. Given $q\in\set{0,1,\ldots,n-1}$, let $\Box^{(q)}_{b,k}$ be the Gaffney extension of Kohn Laplacian for $(0,q)$ forms with values i…

It is shown how the coherent states permit to find different geometrical objects as the geodesics, the conjugate locus, the cut locus, the Calabi's diastasis and its domain of definition, the Euler-Poincaré characteristic, the number of Borel-Morse cells, the Kodaira embedding theorem.

Defines and classifies Thurston geometries and connects simplicial volume to Kodaira dimension.

The abstract discusses embedding theorems for pseudo-Kähler manifolds.

Extends Kodaira Spencer to higher-dimensional Calabi-Yau manifolds.

Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.

Study on Kodaira dimension of SU(m)-structures on almost complex manifolds.

Let $\hat{L}$ be the projective completion of an ample line bundle $L$ over $D$, a smooth projective manifold. Hwang-Singer \cite{HwangS} have constructed complete CSCK metric on $\hat{L}\backslash D$. When the corresponding \kahler form is in the cohomology class of a rational divisor $A$ and when $L$ has negative CSC…

Automorphisms of Kodaira surfaces are shown to be affine transformations.

In this note we show that the Lagrangian Luttinger surgery preserves the symplectic Kodaira dimension. Some constraints on Lagrangian tori in symplectic four manifolds with non-positive Kodaira dimension are also derived.

Computational techniques calculate dimensions of complex structures.

Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.

Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.

We bound the index of a subgroup in iterated Kodaira fibrations.

New findings on Frobenius structures on Kodaira manifolds.

Generalizes Kodaira vanishing theorem to Kahler Lie algebroids.

We define the Kodaira dimension for $3$-dimensional manifolds through Thurston's eight geometries, along with a classification in terms of this Kodaira dimension. We show this is compatible with other existing Kodaira dimensions and the partial order defined by non-zero degree maps. For higher dimensions, we explore th…

Expected centre of mass for random embeddings is constant.

The notion of Kodaira dimension has recently been extended to general almost complex manifolds. In this paper we focus on the Kodaira dimension of almost Kähler manifolds, providing an explicit computation for a family of almost Kähler threefolds on the differentiable manifold underlying a Nakamura manifold. We concent…

The paper explores Kodaira dimension on almost complex manifolds.

The paper uses non-abelian Hodge theory to generalize Kodaira vanishing theorems.

The article provides obstructions for exact submanifolds in symplectic applications.

We provide infinitely many examples of pairs of diffeomorphic, non simply connected K\" ahler manifolds of complex dimension three with different Kodaira dimensions. Also, in any possible Kodaira dimension we find infinitely many pairs of non deformation equivalent, diffeomorphic K\" ahler threefolds.

The paper examines Kodaira dimensions of specific complex 4-manifolds with torsion first Chern class.

Study on Kodaira dimension of specific solvmanifolds without complex structures.

Abstract: Almost Kähler Hodge numbers vary with metric choices.

Analyzes Kodaira-Iitaka dimension and multiplicity using intersection theory.

The paper explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.

A Kodaira fibration is a compact, complex surface admitting a holomorphic submersion onto a complex curve, such that the fibers have nonconstant moduli. We consider Kodaira fibrations X with nontrivial invariant rational cohomology in degree 1, proving that if the dimension of the holomorphic invariants is 1 or 2, then…

We give a direct proof for the asymptotic faithfulness of the quantum $SU(n)$ representations of the mapping class groups using peak sections in Kodaira embedding. We give also estimates on the norm of the parallell transport of the projective connection on the Verlinde bundle. The faithfulness has been proved earlier …

Study of section conjecture analogues over complex numbers and Kodaira fibrations.

Paper extends Kodaira dimension's role in Yamabe invariant for most complex surfaces.

Study on the Kodaira dimension of real parallelizable manifolds with almost complex structures.

The coherent states are viewed as a powerful tool in differential geometry. It is shown that some objects in differential geometry can be expressed using quantities which appear in the construction of the coherent states. The following subjects are discussed via the coherent states: the geodesics, the conjugate locus a…

This is a survey on the various notions of Kodaira dimension in low dimensional topology. The focus is on progress after the 2006 survey [78].

A locally conformally Kaehler (LCK) manifold is a complex manifold admitting a Kaehler covering M, with monodromy acting on M by Kaehler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M. We prove a non-Kaehler analogue of Kodaira embedding theorem: an…

Yamabe invariants of certain non-Kähler surfaces are zero.