arXiv research

Study on Vaisman metrics on Kodaira-Thurston surface using pluriclosed flow.

The abstract discusses transforming Vaisman metrics into Kähler-Einstein structures.

We prove the non-existence of Vaisman metrics on some solvmanifolds with left-invariant complex structures. By this theorem, we show that Oeljeklaus-Toma manifolds does not admit Vaisman metrics.

We prove the non-existence of Vaisman metrics on some solvmanifolds with a left-invariant complex structure. By this theorem, we show that every Oeljeklaus-Toma manifold with $(s,1)$ admits no Vaisman metric.

Extended Vaisman theorem to compact spaces with singularities.

Extends Tian theorem to Vaisman manifolds for approximations.

Study categorizes Vaisman manifolds with vanishing first Chern class and finds canonical metrics.

The pluriclosed flow preserves Vaisman condition on compact complex surfaces.

New proof shows compact homogeneous LCK manifolds are Vaisman.

Study characterizes cohomology of Vaisman manifolds, linking Bott-Chern and Dolbeault numbers.

Study shows compact Vaisman manifolds cannot have certain special Hermitian metrics.

Study on Calabi-Yau locally conformally Kähler manifolds proving they are Vaisman.

Vaisman's theorem extended to locally reducible Kähler spaces.

We extend to metric compact mapping tori a splitting result for coKähler manifolds. In particular, we prove that a compact Vaisman manifold is finitely covered by the product of a Sasakian manifold and a circle.

We give a complete description of all locally conformally Kähler structures with holomorphic Lee vector field on a compact complex manifold of Vaisman type. This provides in particular examples of such structures whose Lee vector field is not homothetic to the Lee vector field of a Vaisman structure. More generally, dr…

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.) mani…

Calabi-Yau theorem extended to Vaisman manifolds.

The metric algebroid proposed by Vaisman (the Vaisman algebroid) governs the gauge symmetry algebra generated by the C-bracket in double field theory (DFT). We show that the Vaisman algebroid is obtained by an analogue of the Drinfel'd double of Lie algebroids. Based on a geometric realization of doubled space-time as …

We prove that a compact lcK manifold with holomorphic Lee vector field is Vaisman provided that either the Lee field has constant norm or the metric is Gauduchon (i.e., the Lee field is divergence-free). We also give examples of compact lcK manifolds with holomorphic Lee vector field which are not Vaisman.

The last years have seen striking improvements on Vaisman's question about existence of locally conformally Kähler (lcK) metrics on compact complex surfaces. The aim of this paper is two-fold. We review results of different authors which, for all known examples of compact complex surfaces, give a complete answer to Vai…

The study counts closed elliptic curves and Reeb orbits on Vaisman and Sasakian manifolds.

Study shows certain Lie groups lead to Oeljeklaus-Toma manifolds.

Proves Vaisman solvmanifolds are finite quotients of Kodaira-Thurston manifolds.

We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this reduction and Kaehler reduction of the universal cover. By a recent result of Kamis…

Vaisman manifolds are strongly related to Kähler and Sasaki geometry. In this paper we introduce toric Vaisman structures and show that this relationship still holds in the toric context. It is known that the so-called minimal covering of a Vaisman manifold is the Riemannian cone over a Sasaki manifold. We show that if…

A locally conformally Kahler manifold is a Hermitian manifold $(M,I,ω)$ satisfying $dω=θ\wedge ω$, where $θ$ is a closed 1-form, called the Lee form of $M$. It is called pluricanonical if $\nablaθ$ is of Hodge type $(2,0)+(0,2)$, where $\nabla$ is the Levi-Civita connection, and Vaisman if $\nablaθ=0$. We show that a c…

Holonomy group of Bismut connection on Vaisman manifolds is studied.

Study on non-Kähler Calabi-Yau manifolds and their geometric structures.

Study of unimodular Sasaki and Vaisman Lie groups, determining all modifications explicitly.

We discuss a correspondence between certain contact pairs on the one hand, and certain locally conformally symplectic forms on the other. In particular, we characterize these structures through suspensions of contactomorphisms. If the contact pair is endowed with a normal metric, then the corresponding lcs form is loca…

Holomorphic tensors on Vaisman manifolds are invariant under the Lee field.

Constructs deformations of Vaisman manifolds preserving foliations.

We characterize unimodular solvable Lie algebras with Vaisman structures in terms of Kähler flat Lie algebras equipped with a suitable derivation. Using this characterization we obtain algebraic restrictions for the existence of Vaisman structures and we establish some relations with other geometric notions, such as Sa…

Study Hermitian metrics with Bismut connection satisfying Bianchi identity and SKT condition.

A Vaisman manifold is a special kind of locally conformally Kaehler 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 manifods of unimodular Lie groups, up to holomorphic isometry. For the case of unimodular Lie gr…

We introduce a class of hermitian metrics with {\em Lee potential}, that generalize the notion of l.c.K. metrics with potential introduced in \cite{ov} and show that in the classical examples of Calabi and Eckmann of complex structures on $S^{2p+1}\x S^{2q+1}$, the corresponding hermitian metrics are of this type. Thes…

We prove that the normal metric contact pairs with orthogonal characteristic foliations, which are either Bochner flat or locally conformally flat, are locally isometric to the Hopf manifolds. As a corollary we obtain the classification of locally conformally flat and Bochner-flat non-Kähler Vaisman manifolds.

New classification for Vaisman manifolds with specific properties.

Integrates C-bracket and Vaisman algebroid structures in double field theory.

A locally conformally Kaehler (l.c.K.) manifold is a complex manifold admitting a Kaehler covering $\tilde M$, with each deck transformation acting by Kaehler homotheties. A compact l.c.K. manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on $\tilde M$. We prove a structure theorem f…

We prove that if a compact nilmanifold $Γ\backslash G$ is endowed with a Vaisman structure, then $G$ is isomorphic to the Cartesian product of the Heisenberg group with $\mathbb{R}$.

A metric is formal if all products of harmonic forms are again harmonic. The existence of a formal metric implies Sullivan formality of the manifold, and hence formal metrics can exist only in presence of a very restricted topology. We show that a warped product metric is formal if and only if the warping function is c…

Sasakian manifolds are odd-dimensional counterpart to Kahler manifolds. They can be defined as contact manifolds equipped with an invariant Kahler 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 o…

For elliptic principal bundles $π:X\ra B$ over Kähler manifolds it was shown by Blanchard that $X$ has a Kähler metric if and only both Chern classes (with real coefficients) of $π$ vanish. For some elliptic principal bundles, when the span of these Chern classes is 1-dimensional, it was shown by Vaisman that $X$ carry…

A locally conformally Kaehler (LCK) manifold is a complex manifold admitting a Kaehler covering M, with monodromy acting on M by Kaehler homotheties. A compact LCK manifold is Vaisman if it admits a holomorphic flow acting by non-trivial homotheties on M. We prove a non-Kaehler analogue of Kodaira embedding theorem: an…

In this paper we investigate the spectral sequence associated to a Riemannian foliation which arises naturally on a Vaisman manifold. Using the Betti numbers of the underlying manifold we establish a lower bound for the dimension of some terms of this cohomological object. This way we obtain cohomological obstructions …

A locally conformally Kähler (LCK) manifold $M$ is one which is covered by a Kähler manifold $\tilde M$ with the deck transform group acting conformally on $\tilde M$. If $M$ admits a holomorphic flow, acting on $\tilde M$ conformally, it is called a Vaisman manifold. Neither the class of LCK manifolds nor that of Vais…

The paper confirms a conjecture for Bismut torsion parallel metrics.