Hermitian Manifolds Research Articles

Hermitian metrics with vanishing second Chern Ricci curvature

AbstractWe describe a rigidity phenomenon exhibited by the second Chern Ricci curvature of a Hermitian metric on a compact complex manifold. This yields a characterisation of second Chern Ricci‐flat Hermitian metrics on several types of manifolds as well as a range of non‐existence results for such metrics.

Frölicher spectral sequence of compact complex manifolds with special Hermitian metrics

In this paper we focus on the interplay between the behaviour of the Frölicher spectral sequence and the existence of special Hermitian metrics on the manifold, such as balanced, SKT or generalized Gauduchon. The study of balanced metrics on nilmanifolds endowed with strongly non-nilpotent complex structures allows us to provide infinite families of compact balanced manifolds with Frölicher spectral sequence not degenerating at the second page. Moreover, this result is extended to non-degeneration at any arbitrary page. Similar results are obtained for the Frölicher spectral sequence of compact generalized Gauduchon manifolds. We also find a compact SKT manifold whose Frölicher spectral sequence does not degenerate at the second page, thus providing a counterexample to a conjecture by Popovici.

Pointwise semi-slant Riemannian maps into almost Hermitian manifolds and Casorati inequalities

UDC 514 As a natural generalization of slant submanifolds [B.-Y. Chen, <em>Bull. Austral. Math. Soc.,</em> <strong>41</strong>, No. 1, 135 (1990)], slant submersions [B. Şahin, <em>Bull. Math. Soc. Sci. Math. Roumanie (N.S.),</em> <strong>54</strong>, No. 102, 93 (2011)], slant Riemannian maps [B. Şahin, <em>Quaestion. Math.,</em> <strong>36</strong>, No. 3, 449 (2013) and Int. J. <em>Geom. Methods Mod. Phys.,</em> <strong>10</strong>, Article 1250080 (2013)], pointwise slant submanifolds [B.-Y. Chen, O. J. Garay, <em>Turk. J. Math.,</em> <strong>36</strong>, 630 (2012)], pointwise slant submersions [J. W. Lee, B. Şahin, <em>Bull. Korean Math. Soc.,</em> <strong>51</strong>, No. 4, 1115 (2014)], pointwise slant Riemannian maps [Y. Gündüzalp, M. A. Akyol, <em>J. Geom. and Phys.,</em> <strong>179</strong>, Article 104589 (2022)], semi-slant submanifolds [N. Papaghiuc, <em>Ann. Ştiinƫ. Univ. Al. I. Cuza Iaṣi. Mat. (N.S.),</em> <strong>40</strong>, 55 (1994)], semi-slant submersions [K.-S. Park, R. Prasad, <em>Bull. Korean Math. Soc.,</em> <strong>50</strong>, No. 3, Article 951962 (2013)], and semi-slant Riemannian maps [K.-S. Park, B. Şahin, <em>Czechoslovak Math. J.,</em> <strong>64</strong>, No. 4, 1045 (2014)], we introduce a new class of Riemannian maps, which are called {\it pointwise semi-slant Riemannian maps,} from Riemannian manifolds to almost Hermitian manifolds. We first give some examples, present a characterization, and obtain the geometry of foliations in terms of the distributions involved in the definition of these maps. We also establish necessary and sufficient conditions for pointwise semi-slant Riemannian maps to be totally geodesic and harmonic, respectively. Finally, we determine the Casorati curvatures for pointwise semi-slant Riemannian maps in the complex space form.

On a parabolic Monge–Ampère type equation on compact almost Hermitian manifolds

AbstractWe investigate a parabolic Monge–Ampère type equation on compact almost Hermitian manifolds and derive a priori gradient and second‐order derivative estimates for solutions to this parabolic equation. These a priori estimates give us higher order estimates and a long‐time solution. Then, we can observe its behavior as .

Quarter-symmetric connection on an almost Hermitian manifold and on a Kähler manifold

The paper observes an almost Hermitian manifold as an example of a generalized Riemannian manifold and examines the application of a quarter-symmetric connection on the almost Hermitian manifold. The almost Hermitian manifold with quarter-symmetric connection preserving the generalized Riemannian metric is actually the Kähler manifold. Observing the six linearly independent curvature tensors with respect to the quarter-symmetric connection, we construct tensors that do not depend on the quarter-symmetric connection generator. One of them coincides with the Weyl projective curvature tensor of symmetric metric $g$. Also, we obtain the relations between the Weyl projective curvature tensor and the holomorphically projective curvature tensor. Moreover, we examine the properties of curvature tensors when some tensors are hybrid.

Pointwise hemi-slant Riemannian maps ($\mathcal{PHSRM}$) from almost Hermitian manifolds

In 2022, the notion of pointwise slant Riemannian maps were introduced by Y. G\"{u}nd\"{u}zalp and M. A. Akyol [Journal of Geometry and Physics, 179, 104589, 2022] as a natural generalization of slant Riemannian maps \cite{s3}, slant Riemannian submersions \cite{lee}, slant submanifolds \cite{c1}. As a generalization of pointwise slant Riemannian maps and many subclasses notions (see also: \cite{ga2}), we introduce {\textit pointwise hemi-slant Riemannian maps (briefly, $\mathcal{PHSRM}$)} from almost Hermitian manifolds to Riemannian manifolds, giving non-trivial (proper) examples and investigate some properties of the map, we deal with their properties: the J-pluriharmonicity, the J-invariant, and the totally geodesicness of the map. Finally, we study some curvature relations in complex space form, involving Chen inequalities and Casorati curvatures for pointwise hemi-slant Riemannian maps, respectively.

Degenerate Complex Monge–Ampère Equations on Some Compact Hermitian Manifolds

Degenerate Complex Monge–Ampère Equations on Some Compact Hermitian Manifolds

Classical Liouville action and uniformization of orbifold Riemann surfaces

We study the classical Liouville field theory on Riemann surfaces of genus g>1 in the presence of vertex operators associated with branch points of orders mi>1. In order to do so, we will consider the generalized Schottky space Sg,n(m) obtained as a holomorphic fibration over the Schottky space Sg of the (compactified) underlying Riemann surface. The fibers of Sg,n(m)→Sg correspond to configuration spaces of n orbifold points of orders m=(m1,…,mn). Drawing on the previous work of Park [] as well as Takhtajan and Zograf [; L. A. Takhtajan and P. Zograf], we define Hermitian metrics hi for tautological line bundles ℒi over Sg,n(m). These metrics are expressed in terms of the first coefficient of the expansion of covering map J near each singular point on the Schottky domain. Additionally, we define the regularized classical Liouville action Sm using Schottky global coordinates on Riemann orbisurfaces with genus g>1. We demonstrate that exp[Sm/π] serves as a Hermitian metric in the holomorphic Q-line bundle ℒ=⊗i=1nℒi⊗(1−1/mi2) over Sg,n(m). Furthermore, we explicitly compute the first and second variations of the smooth real-valued function 𝒮m=Sm−π∑i=1n(mi−1mi)loghi on the Schottky deformation space Sg,n(m). We establish two key results: (i) 𝒮m generates a combination of accessory and auxiliary parameters, and (ii) −𝒮m acts as a Kähler potential for a specific combination of Weil-Petersson and Takhtajan-Zograf metrics that appear in the local index theorem for orbifold Riemann surfaces [Takhtajan and Zograf, ]. The obtained results can then be interpreted in terms of the complex geometry of the Hodge line bundle equipped with Quillen’s metric over the moduli space Mg,n(m) of Riemann orbisurfaces and the tree-level approximation of conformal Ward identities associated with quantum Liouville theory. Published by the American Physical Society 2024

The L2 Aeppli-Bott-Chern Hilbert complex

We analyse the L2 Hilbert complexes naturally associated to a non-compact complex manifold, namely the ones which originate from the Dolbeault and the Aeppli-Bott-Chern complexes. In particular we define the L2 Aeppli-Bott-Chern Hilbert complex and examine its main properties on general Hermitian manifolds, on complete Kähler manifolds and on Galois coverings of compact complex manifolds. The main results are achieved through the study of self-adjoint extensions of various differential operators whose kernels, on compact Hermitian manifolds, are isomorphic to either Aeppli or Bott-Chern cohomology.

Nonnegative Holomorphic Sectional Curvature on Compact Almost Hermitian Manifolds

Nonnegative Holomorphic Sectional Curvature on Compact Almost Hermitian Manifolds

Homotopy BV-algebras in Hermitian geometry

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative BV∞-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.

Degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and applications

In this paper, we show the existence and uniqueness of bounded solutions of the degenerate complex Monge-Ampère type equations on compact Hermitian manifolds and obtain the asymptotics of these solutions. As applications, we give partial answers to the Tosatti-Weinkove conjecture and Demailly-Păun conjecture.

Characterizations of Griffiths positivity, pluriharmonicity and flatness

Deng-Ning-Wang-Zhou showed that a Hermitian holomorphic vector bundle is Griffiths semi-positive if it satisfies the optimal L2-extension condition. As a generalization, we present a quantitative characterization of Griffiths positivity in terms of certain L2-extension conditions. We also show that a R-valued measurable function is pluriharmonic if and only if it satisfies the equality part of the optimal Lp-extension condition. This answers a conjecture of Inayama affirmatively. Moreover, the flatness of a possibly singular Hermitian metric is also equivalent to the equality part of the optimal Lp-extension condition.

CR Compactification for Asymptotically Locally Complex Hyperbolic Almost Hermitian Manifolds

In this article, we consider a complete, non-compact almost Hermitian manifold whose curvature is asymptotic to that of the complex hyperbolic space. Under natural geometric conditions, we show that such a manifold arises as the interior of a compact almost complex manifold whose boundary is a strictly pseudoconvex CR manifold. Moreover, the geometric structure of the boundary can be recovered by analysing the expansion of the metric near infinity.

On the spaces of $(d + d^{c})$-harmonic forms and $(d + d^{\Lambda})$-harmonic forms on almost Hermitian manifolds and complex surfaces

In this paper, we study the spaces of (d+d^{c}) -harmonic forms and of (d+d^{\Lambda}) -harmonic forms, a natural generalization of the spaces of Bott–Chern harmonic forms (respectively, symplectic harmonic forms) from complex (respectively, symplectic) manifolds to almost Hermitian manifolds. We apply the same techniques to compact complex surfaces, computing their Bott–Chern and Aeppli numbers and their spaces of (d+d^{\Lambda}) -harmonic forms. We give several applications to compact quotients of Lie groups by a lattice.

The Second Hessian Type Equation on Almost Hermitian Manifolds

The Second Hessian Type Equation on Almost Hermitian Manifolds

Asymptotic normality of divisors of random holomorphic sections on non-compact complex manifolds

We prove a central limit theorem for divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.

Riemannian Maps From Kaehler Manifold With Generic Fibers

We study Riemannian maps from almost Hermitian manifolds to Riemannian manifolds for the case when the fibers are genericsubmanifold of the total space. We obtain the integrability conditionsfor the distributions while vertical distribution is always integrable. Wealso study the geometry of the leaves of the distribution which arisefrom such maps, and obtain the necessary and sufficient conditions forthe fibers as well as the total manifold to be generic product manifolds. We, further, obtain the necessary and sufficient condition for aRiemannian maps with generic fibers to be totally geodesic map.

Chern flat, Chern–Ricci flat twisted product in almost Hermitian geometry

Abstract We study Chern flat, Chern–Ricci flat twisted product almost Hermitian manifolds. We extend the twisted product to almost Hermitian manifolds, and give some formulae of the Chern curvature, the Chern–Ricci curvature and the Chern scalar curvature on the twisted product almost Hermitian manifold under the quasi-Kähler condition.

Harmonic Complex Structures and Special Hermitian Metrics on Products of Sasakian Manifolds

Harmonic Complex Structures and Special Hermitian Metrics on Products of Sasakian Manifolds