Hermitian Manifolds Research Articles

Oб одном свойстве W4 -многообразий

The properties of almost Hermitian manifolds belonging to the Gray — Hervella class W4 are considered. The almost Hermitian manifolds of this class were studied by such outstanding geometers like Alfred Gray, Izu Vaisman, and Vadim Feodorovich Kirichenko. Using the Cartan structural equations of an almost contact metric structure induced on an arbitrary oriented hypersurface of a W4-manifold, some results on totally umbilical and totally geodesic hypersurfaces of W4-manifolds are presented. It is proved that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a W4-manifold is either homothetic to a Sasakian structure or cosymplectic. Moreover, the quasi-Sasakian structure is cosymplectic if and only if the hypersurface is a to­tally geodesic submanifold of the considered W4-manifold. From the present result it immediately follows that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a locally confor­mal Kählerian (LCK-) manifold also is either homothetic to a Sasakian structure or cosymplectic.

О 6-мерных AH-подмногообразиях класса W1⊕W2⊕W4 алгебры Кэли

Six-dimensional submanifolds of Cayley algebra equipped with an almost Hermitian structure of class W1 W2 W4 defined by means of three-fold vector cross products are considered. As it is known, the class W1 W2 W4 contains all Kählerian, nearly Kählerian, almost Kählerian, locally conformal Kählerian, quasi-Kählerian and Vaisman — Gray manifolds. The Cartan structural equations of the W1 W2 W4 -structure on such six-dimensional submanifolds of the octave algebra are obtained. A criterion in terms of the configuration tensor for an arbitrary almost Hermitian structure on a six-dimensional submanifold of Cayley algebra to belong to the W1 W2 W4 -class is established. It is proved that if a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the quasi-Sasakian hypersurfaces axiom (i.e. a hypersurface with a quasi-Sasakian structure passes through every point of such submanifold), then it is an almost Kählerian manifold. It is also proved that a six-dimensional W1 W2 W4 -submanifold of Cayley algebra satisfies the eta-quasi-umbilical quasi-Sasakian hypersurfaces axiom, then it is a Kählerian manifold.

Deligne pairings and families of rank one local systems on algebraic curves

For smooth families $\mathcal{X} \to S$ of projective algebraic curves and holomorphic line bundles $\mathcal{L, M} \to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\langle \mathcal{L, M} \rangle \to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\mathcal{L}$ and $\mathcal{M}$. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.

Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics. We prove that a compact locally conformal K\"{a}hler manifold with constant nonpositive holomorphic sectional curvature is K\"{a}hler. We also give examples of complete non-K\"{a}hler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature, and complete non-K\"{a}hler metric with zero holomorphic sectional curvature and nonvanishing curvature tensor.

Hermitian-Einstein Metrics For Higgs Bundles Over Complete Hermitian Manifolds

In this paper, we solve the Dirichlet problem for the Hermitian-Einstein equations on Higgs bundles over compact Hermitian manifolds. Then we prove the existence of the Hermitian-Einstein metrics on Higgs bundles over a class of complete Hermitian manifolds.

Horizontally Conformal Submersions from CR-Submanifolds of Locally Conformal Kähler Manifolds

In this paper, we study horizontally conformal submersions from CR-submanifolds of a locally conformal Kahler manifold onto almost Hermitian manifolds, generalizing the results obtained by Şahin (Kodai Math. J. 31, 2008), for horizontally conformal submersions of CR-submanifolds in Kahler ambient space. In particular, we show that any horizontally homothetic submersion of a CR-submanifold M of a locally conformal Kahler manifold with Lee vector field normal to M is a Riemannian submersion up to a scale. Moreover, we obtain that such a map is harmonic, provided that the CR-submanifold is mixed geodesic.

Two-dimensional Hermitian numerical manifold method

Numerous approaches have been proposed to enhance the accuracy and convergence of the numerical manifold method (NMM) in recent years, but most, if not all, of these approaches cannot ensure C1 continuity. Hermitian interpolation is an effective approach for obtaining high-order approximations. However, the requirement of rectangular meshes hinders the application of this approach in the finite element method. Taking advantage of the freedom in meshing in NMM, Hermitian interpolation is incorporated into NMM to obtain the C1 approximation. In contrast to the common high-order NMM, the Hermitian NMM (HNMM) improves the accuracy and convergence without causing the linear dependence problem. Moreover, the degrees of freedom (DOFs) of the mathematical nodes inside the physical domain have physical meanings, and the strains at nodes can be obtained directly without the need for extra postprocessing. The proposed HNMM is verified by solving numerous benchmark linear elastic problems, and the results are compared against those of linear and cubic Lagrangian NMMs. The numerical solutions for these examples confirm the remarkable superiority of the HNMM over the Lagrangian NMMs in terms of accuracy, convergence and efficiency.

A Dirichlet Problem in Noncommutative Potential Theory

We prove the solvability of a Dirichlet problem for flat hermitian metrics on Hilbert bundles over compact Riemann surfaces with boundary. We also prove a factorization result for flat hermitian metrics on doubly connected domains.

A Parabolic Flow of Almost Balanced Metrics

We define a parabolic flow of almost balanced metrics. We show that the flow has a unique solution on compact almost Hermitian manifolds.

The exterior derivative of the Lee form of almost Hermitian manifolds

The exterior derivative dθ of the Lee form θ of almost Hermitian manifolds is studied. If ω is the Kähler two-form, it is proved that the Rω-component of dθ is always zero. Expressions for the other components, in [λ01,1] and in [[λ2,0]], of dθ are also obtained. They are given in terms of the intrinsic torsion. Likewise, it is described some interrelations between the Lee form and U(n)-components of the Riemannian curvature tensor.

L2 Estimates and Vanishing Theorems for Holomorphic Vector Bundles Equipped with Singular Hermitian Metrics

We investigate singular Hermitian metrics on vector bundles, especially strictly Griffiths positive ones. $L^2$ esitimates and vanishing theorems usually require an assumption that vector bundles are Nakano positive. However there is no general definition of the Nakano positivity in singular settings. In this paper, we show various $L^2$ estimates and vanishing theromes by assuming that the vector bundle is strictly Griffiths positive and the base manifold is projective.

Exponential decay of Bergman kernels on complete Hermitian manifolds with Ricci curvature bounded from below

ABSTRACT Given a smooth positive measure μ on a complete Hermitian manifold with Ricci curvature bounded from below, we prove a pointwise Agmon-type bound for the corresponding Bergman kernel, under rather general conditions involving the coercivity of an associated complex Laplacian on -forms. Thanks to an appropriate version of the Bochner–Kodaira–Nakano basic identity, we can give explicit geometric sufficient conditions for such coercivity to hold. Our results extend several known bounds in the literature to the case in which the manifold is neither assumed to be Kähler nor of ‘bounded geometry’. The key ingredients of our proof are a localization formula for the complex Laplacian (of the kind used in the theory of Schrödinger operators) and a mean value inequality for subsolutions of the heat equation on Riemannian manifolds due to Li, Schoen, and Tam. We also show in an appendix that the ‘twisted basic identities’, e.g. [McNeal JD and Varolin D. estimates for the operator. Bull Math Sci. 2015;5(2):179–249] are standard basic identities with respect to conformally Kähler metrics.

On a mixed Monge–Ampère operator for quasiplurisubharmonic functions with analytic singularities

We consider mixed Monge-Amp\`ere products of quasiplurisubharmonic functions with analytic singularities, and show that such products may be regularized as explicit one parameter limits of mixed Monge-Amp\`ere products of smooth functions, generalizing results of Andersson, B{\l}ocki and the last author in the case of non-mixed Monge-Amp\`ere products. Connections to the theory of residue currents, going back to Coleff-Herrera, Passare and others, play an important role in the proof. As a consequence we get an approximation of Chern and Segre currents of certain singular hermitian metrics on vector bundles by smooth forms in the corresponding Chern and Segre classes.

Remarks on Chern–Einstein Hermitian metrics

We study some basic properties and examples of Hermitian metrics on complex manifolds whose traces of the curvature of the Chern connection are proportional to the metric itself.

Channel estimation by tensor-based noise compensation for relay-assisted multiple-input multiple-output and orthogonal frequency division multiplexing indoor visible light communication system

For channel estimation in the relay-assisted multiple-input multiple-output and orthogonal frequency division multiplexing indoor visible light communication (VLC) system affected by light source, there is a problem that the received signal cannot be separated by simple filtering. To solve this problem, a channel estimation by PARATUCK2-PARAFAC-based noise compensation which introduces tensors to construct the suitable model of VLC is proposed. First, the transmitting pilots for channel estimation and noise estimation are constructed separately with the Hermitian symmetry structure. Then the tensor-based transmission model of the indoor VLC system is established, and the signals for channel and noise estimations are received. Finally, an asymptotic estimation for separating signals from different sources and noise is proposed by PARATUCK2-PARAFAC. Simulation results show that the proposed algorithm provides a more effective scheme for channel estimation comparing with the state-of-the-art algorithms.

Lipschitz Normal Embedding Among Superisolated Singularities

Abstract Any germ of a complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated riemannian metric on the germ. A complex analytic germ is said Lipschitz normally embedded (LNE) if its outer and inner metrics are bilipschitz equivalent. LNE seems to be fairly rare among surface singularities; the only known LNE surface germs outside the trivial case (straight cones) are the minimal singularities. In this paper, we show that a superisolated hypersurface singularity is LNE if and only if its projectivized tangent cone has only ordinary singularities. This provides an infinite family of LNE singularities, which is radically different from the class of minimal singularities.

On Quasi-bi-slant Submersions

As a generalization of hemi-slant submersions and semi-slant submersions, we introduce the notion of quasi-bi-slant submersions from almost Hermitian manifolds onto Riemannian manifolds giving some examples and study such submersions from Kahler manifolds onto Riemannian manifolds. We study the geometry of leaves of distributions which are involved in the definition of the submersion. We also obtain conditions for such submersions to be integrable and totally geodesic. Moreover, we give a characterization theorem for proper quasi-bi-slant submersions with totally umbilical fibers.

Conformal semi-invariant Riemannian maps to Kähler manifolds

As a generalization of CR-submanifolds and semi-invariant Riemannian maps, we introduce conformal semi-invariant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give non-trivial examples, investigate the geometry of foliations, and obtain decomposition theorems by using the existence of conformal Riemannian maps. We also investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal anti-invariant Riemannian maps to be totally geodesic.

Extensions of p-Yosida functions to holomorphic mappings of several complex variables

ABSTRACT Let M be a complete complex Hermitian manifold with metric A holomorphic mapping is called p-Yosida mapping if is bounded above for with where is the mapping from to induced by f. We formulate and prove a criterion of holomorphic mappings belonging to the class of all p-Yosida mappings and its applications to establish two different analogues of Lappan's five-point theorem.

RC-POSITIVITY, VANISHING THEOREMS AND RIGIDITY OF HOLOMORPHIC MAPS

AbstractLet $M$ and $N$ be two compact complex manifolds. We show that if the tautological line bundle ${\mathcal{O}}_{T_{M}^{\ast }}(1)$ is not pseudo-effective and ${\mathcal{O}}_{T_{N}^{\ast }}(1)$ is nef, then there is no non-constant holomorphic map from $M$ to $N$. In particular, we prove that any holomorphic map from a compact complex manifold $M$ with RC-positive tangent bundle to a compact complex manifold $N$ with nef cotangent bundle must be a constant map. As an application, we obtain that there is no non-constant holomorphic map from a compact Hermitian manifold with positive holomorphic sectional curvature to a Hermitian manifold with non-positive holomorphic bisectional curvature.