Kodaira Embedding Theorem Research Articles

A new explicit complete Kähler metric on ℂ n

In 1977, Klembeck presented an explicit Kähler metric of positive holomorphic curvature on [Klembeck P. A complete Kähler metric of positive curvature on . Proc Amer Math Soc. 1977;64(2):313–316]. Later, in 1991, To used Klembeck's metric and proved a non-compact version of the Kodaira embedding theorem [To W-K. Quasi-projective embeddings of noncompact complete Kähler manifolds of positive Ricci curvature and satisfying certain topological conditions. Duke Math J. 1991;63(3):745–789]. Wu H-H and Zheng F. [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] discovered another explicit Kähler metric which is a natural perturbation of Klembeck's example. More examples (implicit) are studied by solutions of certain ODEs by Cao H-D. [Existence of gradient Kähler-Ricci solitons. In: Elliptic and parabolic methods in geometry. Minnesota: 1994. p. 1–16; Limits of solutions to the Kähler-Ricci flow. J Differ Geom. 1997;45:257–272] and Wu-Zheng [Examples of positively curved complete Kähler manifolds. In: Geometry and Analysis. No. 1. Somerville (MA): Int. Press; 2011. p. 517–542. (Adv Lect Math (ALM); 17)] respectively. Motivated by these, we investigate other complete Kähler metrics induced from certain weighted Fock spaces, following a similar scheme. In particular, we prove that these Bergman metrics induced by weighted Fock spaces possess positive holomorphic sectional curvature.

Positivity and the Kodaira embedding theorem

Kodaira embedding theorem provides an effective characterization of projectivity of a Kahler manifold in terms the second cohomology. Recently X. Yang [21] proved that any compact Kahler manifold with positive holomorphic sectional curvature must be projective. This gives a metric criterion of the projectivity in terms of its curvature. In this note, we prove that any compact Kahler manifold with positive 2nd scalar curvature (which is the average of holomorphic sectional curvature over 2-dimensional subspaces of the tangent space) must be projective. In view of generic 2-tori being non-abelian, this new curvature characterization is sharp in certain sense.

Szegő kernels and equivariant embedding theorems for CR 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 Kodaira embedding theorem for CR manifolds. Using similar methods we also establish an analytic proof of an $\mathbb R$-equivariant Boutet de Monvel embedding theorem for strongly pseudoconvex CR manifolds. In particular, we obtain equivariant embedding theorems for irregular Sasakian manifolds. As applications of our results, we obtain Torus equivariant Kodaira and Boutet de Monvel embedding theorems for CR manifolds and Torus equivariant Kodaira embedding theorem for complex manifolds.

The fundamental group, rational connectedness and the positivity of Kähler manifolds

Abstract Firstly, we confirm a conjecture asserting that any compact Kähler manifold N with Ric ⊥ > 0 {\operatorname{Ric}^{\perp}>0} must be simply-connected by applying a new viscosity consideration to Whitney’s comass of ( p , 0 ) {(p,0)} -forms. Secondly we prove the projectivity and the rational connectedness of a Kähler manifold of complex dimension n under the condition Ric k > 0 {\operatorname{Ric}_{k}>0} (for some k ∈ { 1 , … , n } {k\in\{1,\dots,n\}} , with Ric n {\operatorname{Ric}_{n}} being the Ricci curvature), generalizing a well-known result of Campana, and independently of Kollár, Miyaoka and Mori, for the Fano manifolds. The proof utilizes both the above comass consideration and a second variation consideration of [L. Ni and F. Zheng, Positivity and Kodaira embedding theorem, preprint 2020, https://arxiv.org/abs/1804.09696]. Thirdly, motivated by Ric ⊥ {\operatorname{Ric}^{\perp}} and the classical work of Calabi and Vesentini [E. Calabi and E. Vesentini, On compact, locally symmetric Kähler manifolds, Ann. of Math. (2) 71 1960, 472–507], we propose two new curvature notions. The cohomology vanishing H q ⁢ ( N , T ′ ⁢ N ) = { 0 } {H^{q}(N,T^{\prime}N)=\{0\}} for any 1 ≤ q ≤ n {1\leq q\leq n} and a deformation rigidity result are obtained under these new curvature conditions. In particular, they are verified for all classical Kähler C-spaces with b 2 = 1 {b_{2}=1} . The new conditions provide viable candidates for a curvature characterization of homogeneous Kähler manifolds related to a generalized Hartshone conjecture.

A note on the generalization of the Kodaira embedding theorem

In this paper, we generalize the famous Kodaira embedding theorem.

Equivariant Kodaira Embedding for CR Manifolds with Circle Action

We consider a compact CR manifold with a transversal CR locally free circle action endowed with an S1-equivariant 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 embedding theorem.

On Projectively Embeddable Complex-Foliated Structures

An extension of Kodaira's embedding theorem is proved for compact smoothly foliated manifolds with complex leaves. Existence of a leafwise positive line bundle is shown to be sufficient for C^k projective embeddability for all k < \infty .

Projectivity of analytic Hilbert and Kähler quotients

We investigate algebraicity properties of quotients of complex spaces by complex reductive Lie groups G. We obtain a projectivity result for compact momentum map quotients of algebraic G-varieties. Furthermore, we prove equivariant versions of Kodaira's Embedding Theorem and Chow's Theorem relative to an analytic Hilbert quotient. Combining these results we derive an equivariant algebraisation theorem for complex spaces with projective quotient.

Morse–Novikov cohomology of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold admitting a Kähler covering, with the monodromy acting on this covering by holomorphic homotheties. We define three cohomology invariants, the Lee class, the Morse–Novikov class, and the Bott–Chern class, of an LCK-structure. These invariants play together the same role as the Kähler class in Kähler geometry. If these classes coincide for two LCK-structures, the difference between these structures can be expressed by a smooth potential, similar to the Kähler case. We show that the Morse–Novikov class and the Bott–Chern class of a Vaisman manifold vanish. Moreover, for any LCK-structure on a manifold, admitting a Vaisman structure, we prove that its Morse–Novikov class vanishes. We show that a compact LCK-manifold M with vanishing Bott–Chern class admits a holomorphic embedding into a Hopf manifold, if dim C M ⩾ 3 , a result which parallels the Kodaira embedding theorem.

A generalization of Matsushima’s embedding theorem

Let $M$ be a compact complex manifold and let $L \rightarrow M$ be a homorphic line bundle whose curvature form is everywhere of signature $(s_+, s_-)$. Under some conditions on the curvature form of $L$, it is show that $K \otimes L^{\otimes m}$ admits, for some $K$ and sufficiently large $m \in \mathbb{N}$, $C^{\infty}$ sections $t_0, \ldots, t_N$ such that the ratio $(t_0 : \cdots : t_N)$ embeds $M$ holomorphically in $s_+$ variables and antiholomorphically in $s_-$ variables. The result extends the Kodaira's embedding theorem as well as aresult of Matsushima for complex tori.

Effective estimates on the very ampleness of the canonical line bundle of locally Hermitian symmetric spaces

We study the problem about the very ampleness of the canonical line bundle of compact locally Hermitian symmetric manifolds of non-compact type. In particular, we show that any suciently large unramied covering of such manifolds has very ample canonical line bundle, and give estimates on the size of the covering manifold, which is itself a locally Hermitian symmetric manifold, in terms of geometric data such as injectivity radius or degree of coverings. Let L be an ample line bundle on an algebraic manifold M. From Kodaira's Embedding Theorem, we know that mL is very ample if m is suciently large so that the sections of mL give rise to an embedding of M into some projective space. A natural question is on the estimates of such m so that mL is very ample. In particular, one may ask the same question for the canonical line bundle of a manifold with negative rst Chern class, or of general type. Of particular interest among such manifolds is the class of locally Hermitian symmetric spaces. The main purpose of this article is to show that for compact locally Hermitian symmetric manifolds of non-compact type, K is very ample if the injectivity radius of the manifold is greater than some eective constant which can be estimated in terms of some geometric data. Eective estimates for mK = K +( m 1)K with m 2 in terms of injectivity radius have been obtained in (HT) and (Y2). The diculty in the case of m =1c an be seen from the fact that in constructing holomorphic sections from the Bochner- Kodaira technique, one always needK +L for some extra positivity fromL .I n (Y2), it is shown that for a tower of coverings of compact Hermitian locally symmetric manifolds Mi, KMi becomes very ample when i approaches to1. To compensate for the extra positivity required for the Bochner-Kodaira technique, we prove our result by showing that the limit of the Bergman kernel on Mi approaches the corresponding one on the universal covering. However, the limiting process makes the whole approach highly non-eective and requires an innite sequence of normal coverings of the original manifold. The main purpose of this article is to show the very ampleness of KM for any unramied covering M of Mo with injectivity radius or degree of the coverings greater than some eectively estimable constant. For this purpose, an alternate approach using heat kernel estimates combined with Atiyah's

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kahler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kahler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.

The Kodaira Embedding Theorem for Kähler varieties with isolated singularities

The Kodaira Embedding Theorem is extended to Kahler varieties with isolated singularities. UsingL 2 estimates for the bundle-valued\(\bar \partial - operator\), it is shown that a necessary and sufficient condition for a compact normal Kahler variety with isolated singularities to be biholomorphic to a projective-algebraic variety is that the variety admit a holomorphic line bundle that is positive when restricted to the regular part of the variety.

The super GAGA principle and families of super Riemann surfaces

We extend the GAGA principle, the Kodaira embedding theorem, and Chow’s lemma to supergeometry and conclude that families of super Riemann surfaces are locally algebraic.

Projective embeddings of complex supermanifolds

We generalize the Kodaira Embedding Theorem and Chow's Theorem to the context of families of complex supermanifolds. In particular, we show that every family of super Riemann surfaces is a family of projective superalgebraic varieties.

An analytical proof of Kodaira's embedding theorem for Hodge manifolds

An analytical proof of Kodaira's embedding theorem for Hodge manifolds

Criteria for quasi-projectivity

In transcendental algebraic geometry it often happens that one gets complex spaces by analytic operations and one wishes to show that they are actually algebraic. In the case of compact complex manifolds and spaces, one has Kodaira's embedding theorem and Grauert's generalization respectively. Proposition I, which is a natural generalization of Kodaira's theorem, gives differential geometric criteria for the quasi-projectivity of certain non-compact manifolds. One corollary of it that gives its flavour is: Corollary. Let X be Zariski open in a compact complex manifold 2g. Let L be a holomorphic line bundle on X that has a Hermitian structure over X with only L 2 poles at infinity and whose curvature form is the Hermitian form associated to a complete metric on X. Then X is Moisezon and X is quasi-projective with respect to the algebraic structure induced J?om f2. The definition of L 2 poles at infinity is given after Proposition I. It is noteworthy that no upper bounds on the growth of the curvature of the Hermitian structure are required! Section I is devoted to Proposition I and some immediate corollaries. The form of Proposition I was dictated by its application to the study of the quasi-projectivity of the image of the Griffiths-Riemann period mapping that arises in algebraic geometry. One can look at [4-7, 9, 10, 19] for background on the period mapping; the above problem is discussed in [7, pp. 259ff.] and [6 (III), Appendix C]. In Section II it is shown that the period mapping satisfies the conditions of Proposition I when the mapping is proper and has a manifold as an image.

A generalization of Kodaira's embedding theorem

A generalization of Kodaira's embedding theorem