BrainMap

Mr. Liviu Ornea

Professor

Professor - UNIVERSITATEA BUCURESTI

Other affiliations

Researcher - INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE (Romania)

Researcher | Teaching staff

Web of Science ResearcherID: F-1800-2010

Curriculum Vitae (08/01/2019)

Expertise & keywords

Geometry and topology of locally conformally Kaehler manifolds Call name: P 4 - Proiecte de Cercetare Exploratorie, 2020
PN-III-P4-ID-PCE-2020-0025 2021 - 2023

Role in this project:

Coordinating institution: UNIVERSITATEA BUCURESTI

Project partners: UNIVERSITATEA BUCURESTI (RO)

Affiliation:

Project website: http://gta.math.unibuc.ro/pages/gvilcu/GTLCK/index.html

Abstract:

Our project concerns complex and Riemannian properties of locally conformally Kähler (LCK) manifolds. In complex geometry, we intend to: count the elliptic curves on compact Vaisman manifolds; study the analytic invariants of the recently found new class of LCK manifolds with global spherical shell (which are not Vaisman); determine the properties of holomorphic submersions between LCK manifolds, aiming to prove the non-existence of LCK products; determine the subspace of Lee forms in the 1-cohomology of a given compact LCK manifold; classify 3-dimensional LCK manifolds according to their algebraic dimension; extend the theory to singular analytic spaces; study the possibility of coexistence of an LCK metric with other non-Kähler metrics (e.g. balanced, astheno-Kähler etc.) on LCK manifolds, in particular on solvmanifolds; study the existence problem of LCK metrics on Oeljeklaus-Toma (OT) manifolds. In Riemannian geometry, we shall concentrate on variational properties (harmonic maps and morphism, Yang-Mills fields and generalizations) and deformations of the canonical foliation of Vaisman manifolds. Our methods will combine real and complex differential geometry techniques with algebraic geometry techniques and, for OT manifolds, number theoretic ones.

Analytic convexity in higher dimensional complex analysis Call name: P 1 - SP 1.1 - Proiecte de cercetare Postdoctorală
PN-III-P1-1.1-PD-2016-0182 2018 - 2020

Role in this project:

Coordinating institution: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE

Project partners: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE (RO)

Affiliation: INSTITUTUL DE MATEMATICA "SIMION STOILOW" AL ACADEMIEI ROMANE (RO)

Project website: https://sites.google.com/view/achdca/home

Abstract:

The main theme of this project is analytic convexity and the problems we are planning to study involve Stein spaces, holomorphic convexity, and q-convexity.

We want to investigate the relation between q-convexity and cohomologic q-convexity. Our purpose will be to obtain some results concerning cohomologic convexity of sets which are finite intersections of (n-1)-complete open subsets of the Euclidean n-dimensional complex space. We want to give some examples of open sets which might be good candidates for proving that cohomologic q-convexity does not imply q-convexity.

We also intend to study if the existence of a Stein neighborhood basis for a compact subset K of the Euclidean n-dimensional complex space with smooth boundary implies the existence of a Stein neighborhood U of K for which K is O(U)-convex. Moreover, we will study the existence of Stein neighborhood bases for some particular unbounded subsets of the Euclidean n-dimensional complex space.

A third problem of interest for this project is to study if any connected, open subset of the base of a locally trivial holomorphic fiber bundle can be represented as the projection of a Stein open connected subset of the total space of the fiber bundle. The case where the fiber bundle is trivial has been already studied and proved to have positive answer, but the technique does not remain valid for non-trivial fiber bundles.

Topics in locally conformally Kähler geometry Call name: P 4 - Proiecte de Cercetare Exploratorie
PN-III-P4-ID-PCE-2016-0065 2017 - 2019

Role in this project:

Coordinating institution: UNIVERSITATEA BUCURESTI

Project partners: UNIVERSITATEA BUCURESTI (RO)

Affiliation: UNIVERSITATEA BUCURESTI (RO)

Project website: http://gta.math.unibuc.ro/pages/gvilcu/LCK/index.html

Abstract:

Locally conformally Kähler (LCK) geometry is concerned with complex manifolds of complex dimension at least two admitting a Kähler covering with deck transformations acting by holomorphic homotheties with respect to the Kähler metric.

Forgetting the complex structure we obtain Locally Conformally Symplectic (LCS) manifolds. The passage between LCS and LCK settings is both directions.

Almost all known non-Kähler compact surfaces are LCK. In higher dimensions, we have Hopf manifolds, Oeljeklaus-Toma (OT) manifolds, and their complex submanifolds (when they do exist).

Objectives:

1. Classification of surjective holomorphic maps from (compact) LCK manifolds. The total space, together with natural conditions on the fibre, should put restrictions on the Hermitian structure of the base.

2. Proving the non-existence of LCK metrics on OT manifolds with t>1.

3. OT manifolds do not have curves and surfaces, and the LCK ones do not have subvarieties at all. We want to prove that if a compact complex submanifold exists in an OT manifold, it must be an OT manifold itself. The problem can be translated into one concerning holomorphic bundles.

4. Studying holomorphic vector bundles (stability, filtrability) of small rank on OT.

5. Initiating the systematic study of complex LCS manifolds, relations with other geometries.

6. Classification of toric LCS manifolds. We expect a statement of this kind: toric, compact LCS manifolds should be LCK and indeed Vaisman.

7. Study of the spectral sequence associated to the canonical foliation of a Vaisman manifold and computing the dimensions of its terms. This will obstruct a given 2-dimensional foliation on a compact complex manifold to come from a Vaisman structure.

8. Study of indefinite LCK manifolds and of their submanifolds, in particular of semi-Riemannian submersions from indefinite LCK manifolds, with focus on statistical manifolds.

9. Studying particular functionals of Yang-Mills type on LCK and Vaisman manifolds.

Locally conformally Kaehler geometry and related structures Call name: Exploratory Research Projects - PCE-2011 call
PN-II-ID-PCE-2011-3-0118 2012 - 2016

Role in this project:

Coordinating institution: Universitatea din Bucuresti

Project partners: Universitatea din Bucuresti (RO)

Affiliation: Universitatea din Bucuresti (RO)

Project website: http://gta.math.unibuc.ro/vuli/eng.html

Abstract:

Locally conformally Kaehler (LCK) geometry is concerned with complex manifolds of complex dimension at least two admitting a Kaehler covering with deck transformations acting by holomorphic homotheties with respect to the Kaehler metric. It is thus a part of complex differential geometry and can be treated using complex methods, Riemannian methods and algebraic geometry methods. The goal of the project is to push forward the knowledge in LCK geometry. We want to further clarify the differences between LCK and Kaehler, and to apply techniques from LCK geometry to other fields. Among the specific objectives we mention the following: we want to determine if the blow-up along subvarieties preserves the LCK class, we study (holomorphic) bundles over LCK manifolds and/or which carry LCK structures (in particular, elliptic bundles over Kaehler bases), toric LCK manifolds (aiming at a Delzant type theorem), harmonic maps and morphisms in LCK setting, indefinite LCK metrics, applications of LCK results and methods to almost contact metric geometries.

Algebraic and combinatorial tools in topology Call name: Postdoctoral Research Projects - PD-2011 call
PN-II-RU-PD-2011-3-0149 2011 - 2013

Role in this project:

Coordinating institution: Universitatea din Bucuresti

Project partners: Universitatea din Bucuresti (RO)

Affiliation:

Project website: https://dl.dropbox.com/u/109689563/CNCS_grant.html

Abstract:

Our project, “Algebraic and combinatorial tools in topology” focuses on combinatorial determination in the field of complex hyperplane arrangements, and the study of (relative) cohomology jump loci of a space in relation to (partial) formality. My aim is to pursue and extend the main directions of investigation of my thesis. One specific objective is to investigate how various aspects of the topology of the complement of an arrangement are reflected by the intersection lattice. This is a recurent research theme for the field and a general expression for major open problems of this theory, for instance combinatoriality of the Milnor fiber monodromy or of the twisted coefficients cohomology. Using, for certain classes of hyperplane arrangements, techniques introduced by Jambu-Papadima [Topology 1998] and Dimca-Papadima [Annals of Math. 2003] we relate the pattern of intersections of hyperplanes (the combinatorics) to homotopical features of the complement. The methods of homotopical algebra developed by Quillen [Annals of Math.1969] and Sullivan [Publ. IHES 1977]) provide powerful tools in topology and geometry. In this direction, as a second objective, we plan to explore new applications of some recently developed concepts: the relative characteristic and resonance varieties [Dimca-Papadima-Suciu, Duke Math. J. 2009] and partial formality properties [M., J. Pure Appl. Alg., 2010].

FILE DESCRIPTION	DOCUMENT
List of research grants as project coordinator	Download (22.5 kb) 04/04/2015
List of research grants as partner team leader	Download (23.5 kb) 04/04/2015
List of research grants as project coordinator or partner team leader
Significant R&D projects for enterprises, as project manager
R&D activities in enterprises
Peer-review activity for international programs/projects	Download (22 kb) 25/11/2019