Note on Locally Conformal Kähler Surfaces

The purpose of this note is to show that the complex two-dimensional locally conformal Kähler solvmanifold obtained by L. de Andres, Fernandez, Mencia and Cordero is holomorphically homothetic to the Inoue surface equipped with the locally conformal Kähler structure constructed by Tricerri. In order to prove it, we collect several facts related to the existence of locally conformal Kähler structure on compact complex surfaces.     

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.     

Semistable Twisted Holomorphic Chains on Non-Compact Kähler Manifolds

In this paper, the author proves a generalized Donaldson-Uhlenbeck-Yau theorem for twisted holomorphic chain on a non-compact K\"ahler manifold. As an application, the author obtains a Bogomolov type Chern numbers inequality for semistable twisted holomorphic chain.     

A Note on the Moment Map on Compact Kähler Manifolds

We consider compact Kähler manifolds acted on effectively by a connected compact Lie group K of isometries in a Hamiltonian fashion. We prove that the squared moment map ||||² is constant if and only if K is semisimple and the manifold is biholomorphically and K-equivariantly isometric to a product of a flag manifold and a compact Kähler manifold which is acted on trivially by K.     

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).     

Locally conformal Kähler submersions

In this paper we study almost Hermitian submersions with total space a locally conformal Kähler (l.c.K.) manifold, i.e., l.c.K. submersions. We derive necessary and sufficient conditions for the fibers of a l.c.K. submersion to be minimal and for the horizontal distribution to be completely integrable. We give, under certain conditions, some relations between the Betti numbers of the total space and the base space of a l.c.K. submersion and we obtain all the l.c.K. submersions with totally geodesic fibers and total space a particular class of generalized Hopf manifolds.Supported by the Consejería de Educación del Gobierno de Canarias     

Transverse Dimension of R~n Actions on Compact Foliated Manifolds

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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.     

On Kirchberg's Inequality for Compact Kähler Manifolds of Even Complex Dimension

In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold of even complex dimension satisfies the inequality , where by S we denote the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T ²×N, where T ² is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture (cf. [11]), but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg's inequality and to prove the above conjecture in some particular cases.     

Holomorphic Lefschetz Fixed Point Formula for Non-compact Khler Manifolds

The authors obtain a holomorphic Lefschetz fixed point formula for certain non-compact "hyperbolic" Khler manifolds (e.g. Khler hyperbolic manifolds, bounded domains of holomorphy) by using the Bergman kernel. This result generalizes the early work of Donnelly and Fefferman.     

Killing Forms on Quaternion-Kähler Manifolds

We show that every Killing p-form on a compact quaternion manifold has to be parallel for p≥ 2. Mathematics Subject Classifications (2000): Primary 53C55, 58J50.     

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.     

Locally conformal Kähler manifolds with potential

A locally conformally Kähler (LCK) manifold M is one which is covered by a Kähler manifold ${\widetilde M}A locally conformally K?hler (LCK) manifold M is one which is covered by a K?hler manifold [(M)\tilde]{\widetilde M} with the deck transformation group acting conformally on [(M)\tilde]{\widetilde M}. If M admits a holomorphic flow, acting on [(M)\tilde]{\widetilde M} conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vaisman manifolds is stable under small deformations. We define a new class of LCK-manifolds, called LCK manifolds with potential, which is closed under small deformations. All Vaisman manifolds are LCK with potential. We show that an LCK-manifold with potential admits a covering which can be compactified to a Stein variety by adding one point. This is used to show that any LCK manifold M with potential, dim M ≥ 3, can be embedded into a Hopf manifold, thus improving similar results for Vaisman manifolds Ornea and Verbitsky (Math Ann 332:121–143, 2005).     

Bochner-Kodaira Techniques on Khler Finsler Manifolds

A horizontal Hodge Laplacian operator □h is defined for Hermitian holomorphic vector bundles over PTM on Khler Finsler manifold,and the expression of □h is obtained explicitly in terms of horizontal covariant derivatives of the Chern-Finsler connection.The vanishing theorem is obtained by using the α_Hα_H-method on Kahler Finsler manifolds.     

Holomorphic submersions of locally conformally Kähler manifolds

A locally conformally Kähler (LCK) manifold is a complex manifold covered by a Kähler manifold, with the covering group acting by homotheties. We show that if such a compact manifold \(X\) admits a holomorphic submersion with positive-dimensional fibers at least one of which is of Kähler type, then \(X\) is globally conformally Kähler or biholomorphic, up to finite covers, to a small deformation of a Vaisman manifold (i.e., a mapping torus over a circle, with Sasakian fiber). As a consequence, we show that the product of a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.     

Regularity Properties of the Degenerate Monge-Ampère Equations on Compact Khler Manifolds

The author establishes a result concerning the regularity properties of the degenerate complex Monge-Ampère equations on compact Khler manifolds.     

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     

Locally conformal Kähler structures on compact nilmanifolds with left-invariant complex structures

Let (M, g, J) be a compact Hermitian manifold and \(\Omega\) the fundamental 2-form of (g, J). A Hermitian manifold (M, g, J) is called a locally conformal Kähler manifold if there exists a closed 1-form α such that \(d\Omega=\alpha \wedge \Omega\) . The purpose of this paper is to give a completely classification of locally conformal Kähler nilmanifolds with left-invariant complex structures.