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Victor Vuletescu

Publications and source records attributed to Victor Vuletescu.

18 recordsLinked to original sources
arXiv2026

Locally Conformally K\"ahler Manifolds of Algebraic Codimension One

A locally conformally K\"ahler (LCK) manifold is a manifold $M$ which admits a K\"ahler structure on its universal cover $\tilde M$, in such a way that the monodromy acts conformally on $\tilde M$. Let $M$ be an $n$-dimensional compact LCK manifold of algebraic dimension $n-1$. We prove that $M$ is bimeromorphic to the total space of an isotrivial elliptic fibration. Morever, there exists an alteration of $M$ which dominates bimeromorphically a manifold admitting a free action of an elliptic curve.

math.DG↗
arXiv2022

Do products of compact complex manifolds admit LCK metrics?

An LCK (locally conformally Kahler) manifold is a Hermitian manifold which admits a Kahler cover with deck group acting by holomorphic homotheties with respect to the Kahler metric. The product of two LCK manifolds does not have a natural product LCK structure. It is conjectured that a product of two compact complex manifolds is never LCK. We classify all known examples of compact LCK manifolds onto three not exclusive classes: LCK with potential, a class of manifolds we call of Inoue type, and those containing a rational curve. In the present paper, we prove that a product of an LCK manifold and an LCK manifold belonging to one of these three classes does not admit an LCK structure.

math.DG↗
arXiv2022

On Locally Conformally K\" ahler metrics on Oeljeklaus-Toma Manifolds

We show that Oeljeklaus-Toma manifolds $X(K, U)$ where $K$ is a number field of signature $(s, t)$ such that $s\geq 1,$ $t\geq 2$ and $s\geq 2t$ admit no lck metric. Combined with the earlier results by K. Oeljeklaus - M. Toma and A. Dubickas this completely solves the problem of existence of LCK metrics on Oeljeklaus-Toma manifolds.

math.DG↗
arXiv2021

On locally conformally Kähler threefolds with algebraic dimension two

The paper is part of an attempt of understanding non-Kähler threefolds. We start by looking at compact complex non-Kähler threefolds with algebraic dimension two and admitting locally conformally Kähler metrics. Under certain assumptions, we prove that they are blown-up quasi-bundles over a projective surface.

math.DG↗
arXiv2018

Automorphisms of OT manifolds and ray class numbers

We compute the automorphism group of OT manifolds of simple type. We show that the graded pieces under a natural filtration are related to a certain ray class group of the underlying number field. This does not solve the open question whether the geometry of the OT manifold sees the class number directly, but brings us a lot closer to a possible solution.

math.DG↗
arXiv2018

Flat affine subvarieties in Oeljeklaus-Toma manifolds

The Oeljeklaus-Toma (OT-) manifolds are compact, complex, non-Kahler manifolds constructed by Oeljeklaus and Toma, and generalizing the Inoue surfaces. Their construction uses the number-theoretic data: a number field $K$ and a torsion-free subgroup $U$ in the group of units of the ring of integers of $K$, with rank of $U$ equal to the number of real embeddings of $K$. We prove that any complex subvariety of smallest possible positive dimension in an OT-manifold is also flat affine. This is used to show that if all non-trivial elements in $U$ are primitive in $K$, then $X$ contains no proper complex subvarieties.

math.DG↗
arXiv2017

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 $L$.

math.AG↗
arXiv2016

Clifford systems in octonionic geometry

We give an inductive construction for irreducible Clifford systems on Euclidean vector spaces. We then discuss how this notion can be adapted to Riemannian manifolds, and outline some developments in octonionic geometry.

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↗
arXiv2013

LCK metrics on Oeljeklaus-Toma manifolds versus Kronecker's theorem

A locally conformally Kähler (LCK) manifold is a manifold which is covered by a Kähler manifold, with the deck transform group acting by homotheties. We show that the search for LCK metrics on Oeljeklaus-Toma manifolds leads to a (yet another) variation on Kronecker's theorem on units. In turn, this implies that on Oeljeklaus-Toma manifold associated to number fields with $2t$ complex embeddings and $s$ real embeddings with $s<t$ there is no LCK metric.

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↗
arXiv2010

Examples of non-trivial rank in locally conformal Kähler geometry

We consider locally conformal Kaehler geometry as an equivariant, homothetic Kaehler geometry (K,Γ). We show that the de Rham class of the Lee form can be naturally identified with the homomorphism projecting Γto its dilation factors, thus completing the description of locally conformal Kaehler geometry in this equivariant setting. The rank r of a locally conformal Kaehler manifold is the rank of the image of this homomorphism. Using algebraic number theory, we show that r is non-trivial, providing explicit examples of locally conformal Kaehler manifolds with 1<r<b_1. As far as we know, these are the first examples of this kind. Moreover, we prove that locally conformal Kaehler Oeljeklaus-Toma manifolds have either r=b_1 or r=b_1/2.

math.DG↗
arXiv2010

LCK metrics on elliptic principal bundles

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 locally conformally Kähler (LCK, for short) metrics. We show that in the case when the Chern classes are linearly independent, $X$ carries no LCK metric.

math.DG↗
arXiv2009

Blowing-up points on l.c. K. manifolds

It is a classical result, due to F. Tricceri, that the blow-up of a manifold of locally conformally Kähler (l.c.K. for short) type at some point is again of l.c.K. type. However, the proof given in \cite{Tric} is somehow unclear. We give a different argument to prove the result, using "standard tricks" in algebraic geometry

math.DG↗