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Liviu Ornea

Publications and source records attributed to Liviu Ornea.

At least 55 records · Page 3Linked to original sources
arXiv2014

Locally conformally Kahler metrics obtained from pseudoconvex shells

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering M, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if M admits an authomorphic Kahler potential. It is known that in this case it is an algebraic cone, that is, the set of all non-zero vectors in the total space of an anti-ample line bundle over a projective orbifold. We start with an algebraic cone C, and show that the set of Kahler metrics with potential which could arise from an LCK structure is in bijective correspondence with the set of pseudoconvex shells, that is, pseudoconvex hypersurfaces in C meeting each orbit of the associated R-action exactly once. This is used to produce explicit LCK and Vaisman metrics on Hopf manifolds, generalizing earlier work by Gauduchon-Ornea and Kamishima-Ornea.

math.DG↗
arXiv2013

Homogeneous locally conformally Kähler manifolds

It is known that automorphism group $G$ of a compact homogeneous locally conformally Kähler manifold $M=G/H$ has at least a 1-dimensional center. We prove that the center of $G$ is at most 2-dimensional, and that if its dimension is 2, then $M$ is Vaisman and isometric to a mapping torus of an isometry of a homogeneous Sasakian manifold.

math.DG↗
arXiv2013

Spin(9) geometry of the octonionic Hopf fibration

We deal with Riemannian properties of the octonionic Hopf fibration S^{15}-->S^8, in terms of the structure given by its symmetry group Spin(9). In particular, we show that any vertical vector field has at least one zero, thus reproving the non-existence of S^1 subfibrations. We then discuss Spin(9)-structures from a conformal viewpoint and determine the structure of compact locally conformally parallel Spin(9)-manifolds. Eventually, we give a list of examples of locally conformally parallel Spin(9)-manifolds.

math.DG↗
arXiv2012

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 Vaisman manifold (i.e. a mapping torus over a circle, with Sasakian fibre). As a consequence, we show that the product between a compact non-Kähler LCK and a compact Kähler manifold cannot carry a LCK metric.

math.DG↗
arXiv2011

Blow-ups of locally conformally Kahler manifolds

A locally conformally Kahler (LCK) manifold is a manifold which is covered by a Kahler manifold, with the deck transform group acting by homotheties. We show that the blow-up of a compact LCK manifold along a complex submanifold admits an LCK structure if and only if this submanifold is globally conformally Kahler. We also prove that a twistor space (of a compact 4-manifold, a quaternion-Kahler manifold or a Riemannian m anifold) cannot admit an LCK metric, unless it is Kahler.

math.AG↗
arXiv2011

An integral invariant from the view point of locally conformally Kähler geometry

In this paper we study an integral invariant which obstructs the existence on a compact complex manifold of a volume form with the determinant of its Ricci form proportional to itself, in particular obstructs the existence of a Kähler-Einstein metric, and has been studied since 1980's. We study this invariant from the view point of locally conformally Kähler geometry. We first see that we can define an integral invariant for coverings of compact complex manifolds with automorphic volume forms. This situation typically occurs for locally conformally Kähler manifolds. Secondly, we see that this invariant coincides with the former one. We also show that the invariant vanishes for any compact Vaisman manifolds.

math.DG↗
arXiv2011

Oeljeklaus-Toma manifolds admitting no complex subvarieties

The Oeljeklaus-Toma (OT-) manifolds are complex manifolds constructed by Oeljeklaus and Toma from certain number fields, and generalizing the Inoue surfaces $S_m$. On each OT-manifold we construct a holomorphic line bundle with semipositive curvature form and trivial Chern class. Using this form, we prove that the OT-manifolds admitting a locally conformally Kahler structure have no non-trivial complex subvarieties. The proof is based on the Strong Approximation theorem for number fields, which implies that any leaf of the null-foliation of w is Zariski dense.

math.CV↗
arXiv2010

Essential points of conformal vector fields

For a conformal vector field $ξ$ on a Riemannian manifold, we say that a point is essential if there is no local metric in the conformal class for which $ξ$ is Killing. We show that the only essential points are isolated zeros of $ξ$. As an application, we show that every connected component of the zero set of $ξ$ is totally umbilical.

math.DG↗
arXiv2010

Submanifolds in manifolds with metric mixed 3-structures

Mixed 3-structures are odd-dimensional analogues of paraquaternionic structures. They appear naturally on lightlike hypersurfaces of almost paraquaternionic hermitian manifolds. We study invariant and anti-invariant submanifolds in a manifold endowed with a mixed 3-structure and a compatible (semi-Riemannian) metric. Particular attention is given to two cases of ambient space: mixed 3-Sasakian and mixed 3-cosymplectic.

math.DG↗
arXiv2010

A report on locally conformally Kähler manifolds

We present an overview of recent results in locally conformally Kähler geometry, with focus on the topological properties which obstruct the existence of such structures on compact manifolds.

math.DG↗
arXiv2010

Remarks on the product of harmonic forms

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 constant and derive further topological obstructions to the existence of formal metrics. In particular, we determine necessary and sufficicient conditions for a Vaisman metric to be formal.

math.DG↗
arXiv2009

Topology of locally conformally Kahler manifolds with potential

Locally conformally Kahler (LCK) manifolds with potential are those which admit a Kahler covering with a proper, automorphic Kaehler potential. Existence of a potential can be characterized cohomologically as a vanishing of a certain cohomology class, called the Bott-Chern class. Compact LCK manifolds with potential are stable at small deformations and admit holomorphic embeddings into Hopf manifolds. This class strictly includes the Vaisman manifolds. We show that every compact LCK manifold with potential can be deformed into a Vaisman manifold. Therefore, every such manifold is diffeomorphic to a smooth elliptic fibration over a Kahler orbifold. We show that the pluricanonical condition on LCK manifolds introduced by G. Kokarev is equivalent to vanishing of the Bott-Chern class. This gives a simple proof of some of the results on topology of pluricanonical LCK-manifolds, discovered by Kokarev and Kotschick.

math.DG↗
arXiv2009

Transformations of locally conformally Kähler manifolds

We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal

math.DG↗
arXiv2009

Automorphisms of locally conformally Kahler manifolds

A manifold M is locally conformally Kahler (LCK) if it admits a Kahler covering with monodromy acting by holomorphic homotheties. For a compact connected group G acting on an LCK manifold by holomorphic automorphisms, an averaging procedure gives a G-invariant LCK metric. Suppose that U(1) acts on an LCK manifold M by holomorphic isometries, and the lifting of this action to the Kahler cover of M is not isometric. We show that the cover admits an automorphic Kahler potential, and hence can be embedded to a Hopf manifold.

math.DG↗