Vaisman Manifolds Research Articles

A -TYPE CONDITION BEYOND THE KÄHLER REALM

Abstract This paper introduces a generalization of the $dd^c$ -condition for complex manifolds. Like the $dd^c$ -condition, it admits a diverse collection of characterizations, and is hereditary under various geometric constructions. Most notably, it is an open property with respect to small deformations. The condition is satisfied by a wide range of complex manifolds, including all compact complex surfaces, and all compact Vaisman manifolds. We show there are computable invariants of a real homotopy type which in many cases prohibit it from containing any complex manifold satisfying such $dd^c$ -type conditions in low degrees. This gives rise to numerous examples of almost complex manifolds which cannot be homotopy equivalent to any of these complex manifolds.

Plurisigned hermitian metrics

Let ( X , ω ) (X,\omega ) be a compact hermitian manifold of dimension n n . We study the asymptotic behavior of Monge-Ampère volumes ∫ X ( ω + d d c φ ) n \int _X (\omega +dd^c \varphi )^n , when ω + d d c φ \omega +dd^c \varphi varies in the set of hermitian forms that are d d c dd^c -cohomologous to ω \omega . We show that these Monge-Ampère volumes are uniformly bounded if ω \omega is “strongly pluripositive”, and that they are uniformly positive if ω \omega is “strongly plurinegative”. This motivates the study of the existence of such plurisigned hermitian metrics. We analyze several classes of examples (complex parallelisable manifolds, twistor spaces, Vaisman manifolds) admitting such metrics, showing that they cannot coexist. We take a close look at 6 6 -dimensional nilmanifolds which admit a left-invariant complex structure, showing that each of them admit a plurisigned hermitian metric, while only few of them admit a pluriclosed metric. We also study 6 6 -dimensional solvmanifolds with trivial canonical bundle.

Bott-Chern cohomology of compact Vaisman manifolds

We give an explicit description of the Bott-Chern cohomology groups of a compact Vaisman manifold in terms of the basic cohomology. We infer that the Bott-Chern numbers and the Dolbeault numbers of a Vaisman manifold determine each other. On the other hand, we show that the cohomological invariants Δ k \Delta ^k introduced by Angella-Tomassini are unbounded for Vaisman manifolds. Finally, we give a cohomological characterization of the Dolbeault and Bott-Chern formality for Vaisman metrics.

Vaisman manifolds and transversally Kähler–Einstein metrics

We use the transverse Kähler–Ricci flow on the canonical foliation of a closed Vaisman manifold to deform the Vaisman metric into another Vaisman metric with a transverse Kähler–Einstein structure. We also study the main features of such a manifold. Among other results, using techniques from the theory of parabolic equations, we obtain a direct proof for the short-time existence of the solution for transverse Kähler–Ricci flow on Vaisman manifolds, recovering in a particular setting a result of Bedulli et al. (J Geom Anal 28:697–725, 2018), but without employing the Molino structure theorem. Moreover, we investigate Einstein–Weyl structures in the setting of Vaisman manifolds and find their relationship with quasi-Einstein metrics. Some examples are also provided to illustrate the main results.

Deformations of Vaisman manifolds

We construct a type of transverse deformation of a Vaisman manifold, which preserves the canonical foliation. For this construction we only need a basic 1-form with certain properties. We show that such basic 1-forms exist in abundance.

Bismut connection on Vaisman manifolds

The holonomy of the Bismut connection on Vaisman manifolds is studied. We prove that if $$M^{2n}$$ is endowed with a Vaisman structure, then the holonomy group of the Bismut connection is contained in $${\text {U}}(n-1)$$ . We compute explicitly this group for particular types of manifolds, namely, solvmanifolds and some classical Hopf manifolds.

J-trajectories in Vaisman manifolds

We show that J-trajectories in Vaisman manifolds have osculating order at most 4. We determine J-trajectories of constant curvatures in the product spaces of Sasakian manifolds and the real line explicitly.

Supersymmetry and Hodge theory on Sasakian and Vaisman manifolds

Sasakian manifolds are odd-dimensional counterpart to Kähler manifolds. They can be defined as contact manifolds equipped with an invariant Kähler structure on their symplectic cone. The quotient of this cone by the homothety action is a complex manifold called Vaisman. We study harmonic forms and Hodge decomposition on Vaisman and Sasakian manifolds. We construct a Lie superalgebra associated to a Sasakian manifold in the same way as the Kähler supersymmetry algebra is associated to a Kähler manifold. We use this construction to produce a self-contained, coordinate-free proof of the results by Tachibana, Kashiwada and Sato on the decomposition of harmonic forms and cohomology of Sasakian and Vaisman manifolds. In the last section, we compute the supersymmetry algebra of Sasakian manifolds explicitly.

Deformations of Vaisman Manifolds

Deformations of Vaisman Manifolds

Vanishing theorems on complete Riemannian manifold with a parallel 1‐form

AbstractIn this article, we first consider the L2 Morse–Novikov cohomology on a complete Riemannian manifold M equipped with a parallel 1‐form which includes Vaisman manifold. Based on a vanishing theorem of L2 Morse–Novikov cohomology, we prove that the L2‐harmonic forms on M are identically zero.

Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds

A compact complex manifold V is called Vaisman if it admits a Hermitian metric which is conformal to a Kähler one, and a non-isometric conformal action by \(\mathbb {C}\). It is called quasi-regular if the \(\mathbb {C}\)-action has closed orbits. In this case the corresponding leaf space is a projective orbifold, called the quasi-regular quotient of V. It is known that the set of all quasi-regular Vaisman complex structures is dense in the appropriate deformation space. We count the number of closed elliptic curves on a Vaisman manifold, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as a quasi-regular quotient of V. We also give a new proof of a result by Rukimbira showing that the number of Reeb orbits on a Sasakian manifold M is either infinite or equal to the sum of all Betti numbers of a Kähler orbifold obtained as an \(S^1\)-quotient of M.

On the Canonical Foliation of an Indefinite Locally Conformal Kähler Manifold with a Parallel Lee Form

We study the semi-Riemannian geometry of the foliation F of an indefinite locally conformal Kähler (l.c.K.) manifold M, given by the Pfaffian equation ω=0, provided that ∇ω=0 and c=∥ω∥≠0 (ω is the Lee form of M). If M is conformally flat then every leaf of F is shown to be a totally geodesic semi-Riemannian hypersurface in M, and a semi-Riemannian space form of sectional curvature c/4, carrying an indefinite c-Sasakian structure. As a corollary of the result together with a semi-Riemannian version of the de Rham decomposition theorem any geodesically complete, conformally flat, indefinite Vaisman manifold of index 2s, 0<s<n, is locally biholomorphically homothetic to an indefinite complex Hopf manifold CHsn(λ), 0<λ<1, equipped with the indefinite Boothby metric gs,n.

Almost formality of quasi-Sasakian and Vaisman manifolds with applications to nilmanifolds

We provide models that are as close as possible to being formal for a large class of compact manifolds that admit a transversely Kaehler structure, including Vaisman and quasi-Sasakian manifolds. As an application we are able to classify the corresponding nilmanifolds.

A new positivity condition for the curvature of Hermitian manifolds

In this note, we introduce a new type of positivity condition for the curvature of a Hermitian manifold, which generalizes the notion of nonnegative quadratic orthogonal bisectional curvature to the non-Kahler case. We derive a Bochner formula for closed (1, 1)-forms from which this condition appears naturally and prove that if a Hermitian manifold satisfies our positivity condition, then any class $$\alpha \in H^{1, 1}_{BC}(X)$$ can be represented by a closed (1, 1)-form which is parallel with respect to the Bismut connection. Lastly, we show that such a curvature positivity condition holds on certain generalized Hopf manifolds and on certain Vaisman manifolds.

Dolbeault cohomology of compact complex manifolds with an action of a complex Lie group

Let G be a complex Lie group acting on a compact complex Hermitian manifold M by holomorphic isometries. We prove that the induced action on the Dolbeault cohomology and on the Bott–Chern cohomology is trivial. We also apply this result to compute the Dolbeault cohomology of Vaisman manifolds.

HOMOGENEOUS SASAKI AND VAISMAN MANIFOLDS OF UNIMODULAR LIE GROUPS

A Vaisman manifold is a special kind of locally conformally Kähler manifold, which is closely related to a Sasaki manifold. In this paper, we show a basic structure theorem of simply connected homogeneous Sasaki and Vaisman manifolds of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie groups, we obtain a complete classification of simply connected Sasaki and Vaisman unimodular Lie groups, up to modification.

Hard Lefschetz theorem for Vaisman manifolds

We establish a Hard Lefschetz Theorem for the de Rham cohomology of compact Vaisman manifolds. A similar result is proved for the basic cohomology with respect to the Lee vector field. Motivated by these results, we introduce the notions of a Lefschetz and of a basic Lefschetz locally conformal symplectic (l.c.s.) manifold of the first kind. We prove that the two notions are equivalent if there exists a Riemannian metric such that the Lee vector field is unitary and parallel and its metric dual 1 1 -form coincides with the Lee 1 1 -form. Finally, we discuss several examples of compact l.c.s. manifolds of the first kind which do not admit compatible Vaisman metrics.

Locally conformally symplectic convexity

We investigate special lcs and twisted Hamiltonian torus actions on strict lcs manifolds and characterize them geometrically in terms of the minimal presentation. We prove a convexity theorem for the corresponding twisted moment map, establishing thus an analog of the symplectic convexity theorem of Atiyah and Guillemin–Sternberg. We also prove similar results for the symplectic moment map (defined on the minimal presentation) whose image is then a convex cone. In the special case of a compact toric Vaisman manifold, we obtain a structure theorem.

Compact complex manifolds bimeromorphic to locally conformally Kähler manifolds

We study compact complex manifolds bimeromorphic to locally conformally Kahler (LCK) manifolds. This is an analogy of studying a compact complex manifold bimeromorphic to a Kahler manifold. We give a negative answer for a question of Ornea, Verbitsky, Vuletescu by showing that there exists no LCK current on blow ups along a submanifold (dim $$\ge 1$$ ) of Vaisman manifolds. We show that a compact complex manifold with LCK currents satisfying a certain condition can be modified to an LCK manifold. Based on this fact, we define a compact complex manifold with a modification from an LCK manifold as a locally conformally class C (LC class C) manifold. We give examples of LC class C manifolds that are not LCK manifolds. Finally, we show that all LC class C manifolds are locally conformally balanced manifolds.

Weighted Bott–Chern and Dolbeault cohomology for LCK-manifolds with potential

A locally conformally Kahler (LCK) manifold is a complex manifold with a Kahler structure on its covering and the deck transform group acting on it by holomorphic homotheties. One could think of an LCK manifold as of a complex manifold with a Kahler form taking values in a local system $L$, called the conformal weight bundle. The $L$-valued cohomology of $M$ is called Morse-Novikov cohomology. It was conjectured that (just as it happens for Kahler manifolds) the Morse-Novikov complex satisfies the $dd^c$-lemma. If true, it would have far-reaching consequences for the geometry of LCK manifolds. Counterexamples to the Morse-Novikov $dd^c$-lemma on Vaisman manifolds were found by R. Goto. We prove that $dd^c$-lemma is true with coefficients in a sufficiently general power $L^a$ of $L$ on any LCK manifold with potential (this includes Vaisman manifolds). We also prove vanishing of Dolbeault and Bott-Chern cohomology with coefficients in $L^a$. The same arguments are used to prove degeneration of the Dolbeault-Frohlicher spectral sequence with coefficients in any power of $L$.