Positive Holomorphic Bisectional Curvature Research Articles

On the curvature of Einstein–Hermitian surfaces

We give a mathematical exposition of the Page metric, and introduce an efficient coordinate system for it. We carefully examine the submanifolds of the underlying smooth manifold, and show that the Page metric does not have positive holomorphic bisectional curvature. We exhibit a holomorphic subsurface with flat normal bundle. We also give another proof of the fact that a compact complex surface together with an Einstein–Hermitian metric of positive orthogonal bisectional curvature is biholomorphically isometric to the complex projective plane with its Fubini–Study metric up to rescaling. This result relaxes the Kähler condition in Berger’s theorem, and the positivity condition on sectional curvature in a theorem proved by the second author.

THE <TEX>${\bar{\partial}}$</TEX>-PROBLEM WITH SUPPORT CONDITIONS AND PSEUDOCONVEXITY OF GENERAL ORDER IN KÄHLER MANIFOLDS

Let M be an n-dimensional <TEX>$K{\ddot{a}}hler$</TEX> manifold with positive holomorphic bisectional curvature and let <TEX>${\Omega}{\Subset}M$</TEX> be a pseudoconvex domain of order <TEX>$n-q$</TEX>, <TEX>$1{\leq}q{\leq}n$</TEX>, with <TEX>$C^2$</TEX> smooth boundary. Then, we study the (weighted) <TEX>$\bar{\partial}$</TEX>-equation with support conditions in <TEX>${\Omega}$</TEX> and the closed range property of <TEX>${\bar{\partial}}$</TEX> on <TEX>${\Omega}$</TEX>. Applications to the <TEX>${\bar{\partial}}$</TEX>-closed extensions from the boundary are given. In particular, for q = 1, we prove that there exists a number <TEX>${\ell}_0$</TEX> > 0 such that the <TEX>${\bar{\partial}}$</TEX>-Neumann problem and the Bergman projection are regular in the Sobolev space <TEX>$W^{\ell}({\Omega})$</TEX> for <TEX>${\ell}$</TEX> < <TEX>${\ell}_0$</TEX>.

Rotational symmetry of conical Kähler–Ricci solitons

We show that expanding K\"ahler-Ricci solitons which have positive holomorphic bisectional curvature and are asymptotic to K\"ahler cones at infinity must be the U(n)-rotationally symmetric expanding solitons constructed by Cao.

Holomorphic curvatures of twistor spaces

We study the twistor spaces of oriented Riemannian 4-manifolds as a source of almost Hermitian 6-manifolds of constant or strictly positive holomorphic, Hermitian and orthogonal bisectional curvatures. In particular, we obtain explicit formulas for these curvatures in the case when the base manifold is Einstein and self-dual, and observe that the "squashed" metric on ℂℙ3 is a non-Kähler Hermitian–Einstein metric of positive holomorphic bisectional curvature. This shows that a recent result of Kalafat and Koca [M. Kalafat and C. Koca, Einstein–Hermitian 4-manifolds of positive bisectional curvature, preprint (2012), arXiv: 1206.3941v1 [math.DG]] in dimension four cannot be extended to higher dimensions. We prove that the Hermitian bisectional curvature of a non-Kähler Hermitian manifold is never a nonzero constant which gives a partial negative answer to a question of Balas and Gauduchon [A. Balas and P. Gauduchon, Any Hermitian metric of constant non-positive (Hermitian) holomorphic sectional curvature on a compact complex surface is Kähler, Math. Z.190 (1985) 39–43]. Finally, motivated by an integrability result of Vezzoni [L. Vezzoni, On the Hermitian curvature of symplectic manifolds, Adv. Geom.7 (2007) 207–214] for almost Kähler manifolds, we study the problem when the holomorphic and the Hermitian bisectional curvatures of an almost Hermitian manifold coincide. We extend the result of Vezzoni to a more general class of almost Hermitian manifolds and describe the twistor spaces having this curvature property.

On L 2-Estimates for $\bar{\partial}$ on a Pseudoconvex Domain in a Complete Kähler Manifold with Positive Holomorphic Bisectional Curvature

Let Ω be a relatively compact pseudoconvex domain in a complete Kahler manifold X with positive holomorphic bisectional curvature. If Ω has positive inner reach and is defined by a plurisubharmonic function of class \(\mathcal{C}^{1}\), we generalize the existence of the Diederich–Fornaess exponent for the distance function to the boundary ∂Ω. This property allows us to prove L2 estimates for the \(\bar{\partial}\) operator and regularity properties for the \(\bar{\partial}\)-Neumann operator.

Twistor Spaces with Positive Holomorphic Bisectional Curvature

In this paper we provide twistorial examples of compact Hermitian manifolds with positive holomorphic bisectional curvature. We also observe that the so-called “squashed” metric on CP 3 , the twistor space of the sphere S 4 ,

Estimées [formula omitted] pour lʼopérateur [formula omitted] sur des domaines pseudoconvexes dans une variété kählerienne complète à courbure bisectionnelle holomorphe strictement positive

We study L2-estimates for ∂¯ on pseudoconvex domains Ω relatively compact in a complete Kähler manifold with positive holomorphic bisectional curvature. We define a function τΩ on C2-smooth pseudoconvex domains. On these domains, we prove that there exists a neighborhood of ∂Ω on which τΩ<1. A generalization of this property allows us, among others, to prove L2-estimates for ∂¯ on C1 pseudoconvex domains Ω with a defining plurisubharmonic function and a positive inner reach.

A survey of the Kähler-Ricci Flow and Yau’s Uniformization Conjecture

Yau’s uniformization conjecture states: a complete noncompact Kahler manifold with positive holomorphic bisectional curvature is biholomorphic to C. The Kahler-Ricci flow has provided a powerful tool in understanding the conjecture, and has been used to verify the conjecture in several important cases. In this article we present a survey of the Kahler-Ricci flow with focus on its application to uniformization. Other interesting methods and results related to the study of Yau’s conjecture are also discussed.

Stability of gradient Kähler-Ricci solitons

We study stability of non-compact gradient Kahler-Ricci ow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kahler potential of the soliton will con- verge to the original soliton under Kahler-Ricci ow as time tends to innity. To obtain this result, we construct appropriate barriers and introduce an L p -norm that decays for these barriers with non-negative Ricci curvature.

The projective rank of a Hermitian symmetric space: A geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds \(N\subset M,\) having positive holomorphic bisectional curvature with the induced Kahler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, \({\rm Pr} \left( M\right),\) the usual rank, \(rk\left( M\right),\) and the 2-number \(\#_2\left( M\right)\) (which is known to be equal to the Euler-Poincare characteristic in these spaces).

A uniformization theorem for complete Kähler manifolds with positive holomorphic bisectional curvature

In the theory of complex geometry, one of the famous problems is the following conjecture of Greene and Wu [13] and Yau [33]: Suppose M is a complete noncompact Kahler manifold with positive holomorphic bisectional curvature; then M is biholomorphic to ℂn. In this paper we use the Ricci flow evolution equation to study this conjecture and prove the result that if M has bounded and positive curvature such that the L’ norm of the curvature on geodesic ball is small enough, then the conjecture is true. Our result gives an improvement on the results of Mok et al. [21] and Mok [22].

On the normal bundle of a complex submanifold of a locally conformal Kaehler manifold

We investigate the existence of parallel sections in the normal bundle of a complex submanifold of a locally conformal Kaehler manifold with positive holomorphic bisectional curvature. Also, if∼M is a quasi-Einstein generalized Hopf manifold then we show that any complex submanifoldM with a flat normal connection of∼M is quasi-Einstein, too, provided thatM is tangent to the Lee field of∼M. As an application of our results we study the geometry of the second fundamental form of a complex submanifold in the locally conformal Kaehler sphereQ m (of a complex Hopf manifoldS2m+1 ×S1).

Projective manifolds whose tangent bundles are numerically effective

In 1979, Mori [Mo] proved the so-called Hartshorne-Frankel conjecture: Every projective n-dimensional manifold with ample tangent bundle is isomorphic to the complex projective space P,. A differential-geometric analogon assuming the existence of a K/ihler metric on X with positive holomorphic bisectional curvature is independently due to Siu-Yau [SY]. Thus it seems natural to classify projective manifolds X whose tangent bundle Tx satisfy a degenerate condition of ampleness: numerical effectivity (abbreviated by "nef'). This means that the tautological quotient line bundle d~(1) on F(Tx) is numerically effective, i.e. C_>_0

Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature

Complete noncompact Kähler manifolds with positive holomorphic bisectional curvature

A Compact Kähler Surface of Negative Curvature Not Covered by the Ball

The uniformization theorem for Riemann surfaces says that a simply connected Riemann surface must be the Riemann sphere, the whole complex plane, or the open unit disc. In the higher dimensional case there is no such simple trichotomy, because generic slight perturbations of the ball give rise to complex manifolds no two of which are biholomorphic [1]. However, it is conjectured that with reasonable curvature assumptions a similar trichotomy exists in the higher dimensional case. Corresponding to the case of the Riemann sphere, one has the Frankel conjecture that a compact Kihler manifold of positive sectional curvature must be biholomorphic to the complex projective space. This conjecture was proved in the dimension 2 case by Andreotti-Frankel [2] and in the dimension 3 case by Mabuchi [5]. Very recently the general case was proved independently by Mori [6] using algebraic geometry of positive characteristic and by Siu-Yau [11] using the complex-analyticity of harmonic maps. (Mori's result is stronger than the result of Siu-Yau. Mori's result assumes only that the manifold has ample tangent bundle, whereas the result of Siu-Yau assumes that the manifold has positive holomorphic bisectional curvature.) Corresponding to the case of the complex plane, one has the conjecture that a noncompact complete Kahler manifold of positive sectional curvature must be biholomorphic to some C. Or, more generally, a noncompact simply connected complete Kahler manifold with sectional curvature K > -A/r'+ or even with the weaker assumption K ? -lk(r) with k(r) > 0 and

Curvature characterization of hyperquadrics

After proving the dimension two case jointly with Andreotti, Frankel [3] conjectured that a compact K/ihler manifold of positive sectional curvature is biholomorphic to the complex projective space. Mabuchi [8] verified the case of dimension three by using the result of Kobayashi-Ochiai [6]. Very recently by using the methods of algebraic geometry of positive characteristic Mori [10] proved that a compact Kihler manifold with ample tangent bundle must be biholomorphic to the complex projective space. By methods of Kihler geometry Siu-Yau [12] proved that a compact K/ihler manifold of positive holomorphic bisectional curvature must be biholomorphic to the complex projective space. Frankel’s conjecture is a special case of these more general results. It is reasonable to conjecture that there are similar curvature characterizations for other irreducible compact symmetric K/ihler manifolds. In this paper we obtain such a curvature characterization for the complex hyperquadric. Definition. Let M be a K/ihler manifold and P M. If the holomorphic bisectional curvature of M is nonnegative at P, then for a nonzero element (a) of the holomorphic tangent space T,M of M at P, the curvature null space at P in the direction of , denoted by N,(O, is defined as the set of all

Some applications of Synge’s formula to the theory of several complex variables

In [10] and [11], the second author proved the following theorem by using Synge’s formula:THEOREM I. Let M be a kähler manifold with positive holomorphic bi-sectional curvature. Then every pseudoconvex domain in M is a Stein manifold.

Supplement to Pseudoconvex Domains on a Kähler Manifold with Positive Holomorphic Bisectional Curvature

In O. Suzuki [3], the author proved that if M has a real analytic kahler metric with positive holomorphic bisectional curvature, then every relatively compact pseudoconvex domain in M is holomorphically convex. After the completion of O. Suzuki [3], the paper of G. Elencwajg [1] appeared. There he proved the same result as in O. Suzuki [3] in the case of kahler metrics of C°°-class. Therefore we see that both results

Pseudoconvex Domains on a Kahler Manifold with Positive Holomorphic Bisectional Curvature

Let M be an ^-dimensional complex manifold with a real analytic kahler metric. Throughout this paper, a kahler manifold is assumed to have a real analytic kahler metric without mentioning it. A relatively compact domain D in M is called a pseudoconvex (resp. strongly pseudoconvex) domain if there exist a neighborhood U of p and a pseudoconvex (resp. strongly pseudoconvex) function (p on U satisfying Df] U= {(p<$ for each boundary point p^dD. We write simply s-pseudoconvex domains (resp. functions) for strongly pseudoconvex domains (resp. functions). Note that pseudoconvex domains are not always Stein manifolds. The purpose of the present paper is to show the following theorem:

Three-dimensional compact K\"ahler manifolds with positive holomorphic bisectional curvature

Three-dimensional compact K\"ahler manifolds with positive holomorphic bisectional curvature