K?hler Manifolds with Almost Non-negative Ricci Curvature

Compact Kähler manifolds with semi-positive Ricci curvature have been investigated by various authors. From Peternell’s work, if M is a compact Kähler n-manifold with semi-positive Ricci curvature and finite fundamental group, then the universal cover has a decomposition \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{M} \cong X_{1} \times \cdots \times X_{m} \), where X _j is a Calabi-Yau manifold, or a hyperKähler manifold, or X _j satisfies H ⁰(X _j , Ω ^p ) = 0. The purpose of this paper is to generalize this theorem to almost non-negative Ricci curvature Kähler manifolds by using the Gromov-Hausdorff convergence. Let M be a compact complex n-manifold with non-vanishing Euler number. If for any ∈ > 0, there exists a Kähler structure (J _∈, g _∈) on M such that the volume \({\text{Vol}}_{{g_{ \in } }} {\left( M \right)} < V\), the sectional curvature |K(g _∈)| < Λ², and the Ricci-tensor Ric(g _∈)> ?∈g _∈, where V and Λ are two constants independent of ∈. Then the fundamental group of M is finite, and M is diffeomorphic to a complex manifold X such that the universal covering of X has a decomposition, \( \ifmmode\expandafter\tilde\else\expandafter\~\fi{X} \cong X_{1} \times \cdots \times X_{s} \), where X _i is a Calabi-Yau manifold, or a hyperKähler manifold, or X _i satisfies H ⁰(X _i , Ω ^p ) = {0}, p > 0.     

On the Curvature of Kähler Manifolds with Zero Ricci Tensor

Mathematical Notes - The behavior of the modulus of the curvature tensor and of the holomorphic sectional curvature on Ricci-flat Kähler manifolds is investigated.     

Two-Orbit Kähler Manifolds and Morse Theory

We deal with compact Kähler manifolds M acted on by a compact Lie group K of isometries, whose complexification K has exactly one open and one closed orbit in M. If the K-action is Hamiltonian, we investigate topological and cohomological properties of M.     

Quasi—conformally Flat Manifolds with Constant Scalar Curvature

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Flat nearly Kähler manifolds

We classify flat strict nearly Kähler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-Kähler factor of maximal dimension and a strict flat nearly Kähler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form $\zeta \in \Lambda^3 (\mathbb{C}^m)^*We classify flat strict nearly K?hler manifolds with (necessarily) indefinite metric. Any such manifold is locally the product of a flat pseudo-K?hler factor of maximal dimension and a strict flat nearly K?hler manifold of split signature (2m, 2m) with m ≥ 3. Moreover, the geometry of the second factor is encoded in a complex three-form . The first nontrivial example occurs in dimension 4m = 12.      

Constant Scalar Curvature Kähler Surfaces and Parabolic Polystability

A complex ruled surface admits an iterated blow-up encoded by a parabolic structure with rational weights. Under a condition of parabolic stability, one can construct a Kähler metric of constant scalar curvature on the blow-up according to Rollin and Singer (J. Eur. Math. Soc., 2004). We present a generalization of this construction to the case of parabolically polystable ruled surfaces. Thus, we can produce numerous examples of Kähler surfaces of constant scalar curvature with circle or toric symmetry.     

Szasz Analytic Functions and Noncompact Kähler Toric Manifolds

We show that the classical Szasz analytic function S _N (f)(x) is obtained by applying the pseudo-differential operator f(N ^?1 D _θ ) to the Bergman kernels for the Bargmann–Fock space. The expression generalizes immediately to any smooth polarized noncompact complete toric Kähler manifold, defining the generalized Szasz analytic function \(S_{h^{N}}(f)(x)\). About \(S_{h^{N}}(f)(x)\), we prove that it admits complete asymptotics and there exists a universal scaling limit. As an example, we will further compute \(S_{h^{N}}(f)(x)\) for the Bergman metric on the unit ball.     

Compact Almost Kähler Manifolds with Divergence-Free Weyl Conformal Tensor

We prove that a compact almost Kähler manifold satisfying that a certain part of thedivergence W of the Weyl conformal tensor W vanishes isKähler.     

The Khler-Ricci Flow on Khler Manifolds with 2-Non-negative Traceless Bisectional Curvature Operator

The authors show that the 2-non-negative traceless bisectional curvature is preserved along the Khler-Ricci flow. The positivity of Ricci curvature is also preserved along the Khler-Ricci flow with 2-non-negative traceless bisectional curvature. As a corollary, the Khler-Ricci flow with 2-non-negative traceless bisectional curvature will converge to a Khler-Ricci soliton in the sense of Cheeger-Gromov-Hausdorff topology if complex dimension n ≥ 3.     

Almost Flat Theorem for Complete Open Riemannian Manifolds with Nonnegative Ricci Curvature

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Lelong-Skoda Transform for Compact Kähler Manifolds and Self-Intersection Inequalities

Let X be a compact Kähler manifold of dimension k and T be a positive closed current on X of bidimension (p,p) (1≤p<k?1). We construct a continuous linear transform ? _p (T) of T which is a positive closed current on X of bidegree (1,1) which has the same Lelong numbers as T. We deduce from that construction self-intersection inequalities for positive closed currents of any bidegree.     

Real Lightlike Hypersurfaces of Paraquaternionic Kähler Manifolds

The main purpose of this paper is to give basic properties of real lightlike hypersurfaces of paraquaternionic manifold and to prove the nonexistence of real lightlike hypersurfaces in paraquaternionic space forms under some conditions. Dedicated to the memory of Professor Aldo Cossu     

Two-Orbit K?hler Manifolds and Morse Theory

We deal with compact Kähler manifolds M acted on by a compact Lie group K of isometries, whose complexification K has exactly one open and one closed orbit in M. If the K-action is Hamiltonian, we investigate topological and cohomological properties of M.     

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

The main result of this paper is that a connected bounded geometry complete K?hler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson’s groups V, T, and F, are not K?hler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact K?hler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2. Received: January 2006, Revision: November 2006, Accepted: March 2007     

The Geometry of Positive Locally Quaternion Kähler Manifolds

A classification of locally quaternion Kähler manifolds M ⁴ⁿ with positive scalar curvature is obtained as a consequence of J. Wolf's work on space forms of irreducible symmetric spaces. We determine the Betti numbers of such manifolds M ⁴ⁿ as well as of the projective 3-Sasakian manifolds fibering over them. We study the geometry of the quaternion Kähler and locally quaternion Kähler submanifolds for each M ⁴ⁿ, which is particularly significant for 4n = 16 due to its relation with four quaternionic structures on the Grassmannian (R ⁸).     

Compatible Almost Complex Structures on Quaternion Kähler Manifolds

Let (M⁴ⁿ,g,Q) be a quaternion Kähler manifold with reduced scalar curvature = K/4n(n + 2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let = – F J be the Lee form of J. We prove the following local results: (1) if J is conformally symplectic, then it is parallel and = 0; (2) if J is cosymplectic, then 0 with equality if and only if J is parallel; (3) if J is integrable, then d is Q-Hermitian and harmonic; and (4) any closed self-dual 2-form = f(g J) ² ₊ = g Q ² is parallel. In Section 5, extending previous results of Salamon [24], we describe a correspondence among conformally balanced J, Killing vector fields X and self-dual 2-forms satisfying the twistor equation.When M⁴ⁿ is compact our main global results are the following: (1) if > 0, then there exists no compatible almost complex structure J; (2) if the first Chern class c₁(T^(1,0) _J M) = 0, then = 0; (3) if = 0 a compatible complex structure J is parallel; and (4) if 0, then no compatible complex structure J exists. The last two results have been proved in [23] by twistor methods.     

Cohomogeneity one Kähler and Kähler–Einstein manifolds with one singular orbit I

Let M be a cohomogeneity one manifold of a compact semisimple Lie group G with one singular orbit \(S_0 = G/H\). Then M is G-diffeomorphic to the total space \(G \times _H V\) of the homogeneous vector bundle over \(S_0\) defined by a sphere transitive representation of G in a vector space V. We describe all such manifolds M which admit an invariant Kähler structure of standard type. This means that the restriction \(\mu : S = Gx = G/L \rightarrow F = G/K \) of the moment map of M to a regular orbit \(S=G/L\) is a holomorphic map of S with the induced CR structure onto a flag manifold \(F = G/K\), where \(K = N_G(L)\), endowed with an invariant complex structure \(J^F\). We describe all such standard Kähler cohomogeneity one manifolds in terms of the painted Dynkin diagram associated with \((F = G/K,J^F)\) and a parameterized interval in some T-Weyl chamber. We determine which of these manifolds admit invariant Kähler–Einstein metrics.     

On Paraquaternionic Submersions Between Paraquaternionic Kähler Manifolds

In this paper we deal with some properties of a class of semi-Riemannian submersions between manifolds endowed with paraquaternionic structures, proving a result of non-existence of paraquaternionic submersions between paraquaternionic Kähler non-locally hyper para-Kähler manifolds. Then we examine, as an example, the canonical projection of the tangent bundle, endowed with the Sasaki metric, of an almost paraquaternionic Hermitian manifold.