Brief description of locally conformally Kähler manifolds

The locally conformally Kähler manifolds are those complex manifolds which have the the property that on their universal cover there exists a Kähler metric upon which the deck transformations act by homotheties. Another (equivalent) definition is: a hermitian manifold X is locally conformally Kähler if the alternate 2-form associated to the metric,
Ω(X, Y)=g(X, JY)
obeys a relation of the type
dΩ=θ ∧ Ω
where θ is a closed 1-form (called the Lee form).
Locally conformally Kähler manifolds have been intensively studied in the last three decades. To-day, the dichotomy between Kähler and respectively locally conformally Kähler manifolds is still not completely understood. For instance, only some restrictions on the fundamental group are known, but the study of this invariant of locally conformally Kähler manifolds is still at his beginning. In particular, an answer to the question whether these manifolds are formal is still unknown.

Examples of locally conformally Kähler manifolds are abundant. The list of these manifolds already contains almost all non-Kähler surfaces, with two exceptions: a class of Inoue surfaces and the possible surfaces in Kodaira's class VII with positive second Betti number and without global spherical shells (a famous conjecture asserts that this last class of surfaces does not even exists).In higher dimensions, there are already two important classes of locally conformally Kahler manifolds, namely the so-called Vaisman manifolds (and their small deformations) and the Oeljeklaus-Toma manifolds.

The Vaisman manifolds are probably one of the most important class of locally conformally Kähler manifolds. They were introduced and studied (in a long series of papers) by I. Vaisman in the '80's. In dimension 2, they are already completely classified by F. Belgun, while in higher dimensions their structure was clarified by L. Ornea and M. Verbitsky; these manifolds are torus fibrations over Sasaki manifolds. A special class of Vaisman manifolds are the locally conformally HyperKähler manifolds: one knows that their study is closely related to the study of HKT manifolds, and are intensively studied in the last decade. This represents a very interesting direction for further investigations.

The Oeljeklaus-Toma manifolds. These non-Kähler manifolds was introduced by K. Oeljeklaus and M. Toma, are associated to number fields and can be seen as generalisations to higher dimensions of a class of Inoue surfaces. Recently, one remarked (Ornea-Verbitsky) that these manifolds have, like the Vaisman ones, a 2-dimensional positive foliation. In particular, using these foliation, one could prove that these manifolds have no proper, closed analytical spaces.

Locally conformally Kähler manifolds with potential. The universal cover of a Vaisman manifold has a Kähler metric with automorphic potential. On the other hand, there are examples of non-Vaisman manifolds which also have this property. In a series of papers with M. Verbistky we clarified the structure of these manifolds. In particular, we know now that these manifolds are deformations of Vaisman manifolds and consequently their topology is completely determined. On the other hand, LCK manifolds with potential admit a characterisation in terms of holomorphic and conformal actions of the unit. More, one can prove that these manifolds admit an embedding in a Hopf manifold, which is an analog of Kodaira's embedding theorem.

Objectives and workplan

Project director:

Prof. Dr. Liviu Ornea, Facultatea de Matematica si Informatica a Universitatii din Bucuresti.

Budget

Reports

2012:- Unique phase
- Deviz postcalcul
- Scientific report

2013:- Unique phase
- Deviz postcalcul
- Scientific report

2014:- Unique phase
- Deviz postcalcul
- Scientific report

2015:- Unique phase
- Deviz postcalcul
- Scientific report

2016:- Unique phase
- Deviz postcalcul
- Scientific report

Research funded by this grant:

Articles published or in press in ISI ranked journals

1. L. Ornea, M. Verbitsky, V. Vuletescu: Blow-ups of Locally Conformally Kahler Manifolds, International Mathematics Research Notices, 12 (2013), 2809-2821.

2. G. E. Vilcu: Ruled CR-submanifolds of locally conformal Kahler manifolds, Journal of Geometry and Physics, 62 (6) (2012), 1366-1372.

3. M Visinescu, G. Vilcu: Hidden symmetries of Euclideanised Kerr-NUT-(A)dS metrics in certain scaling limits, Symmetry, Integrability and Geometry: Methods and Applications (SIGMA) , 8 (2012), 058

4. C. Gherghe: Harmonic maps and stability on locally conformal Kahler manifolds, Journal of Geometry and Physics 70 (2013), 48-53

5. L. Ornea, M. Parton, V. Vuletescu: Holomorphic submersions of locally conformally Kahler manifolds, Annali di Matematica Pura ed Applicata, 193 (5), (2104),1345-1351

6. G. E. Vilcu: Mixed paraquaternionic 3-submersions, Indagationes Mathematicae, 24(2) (2103), 474-488

7. G. E. Vilcu: On Chen invariants and inequalities in quaternionic geometry , Journal of Inequalities and Applications 2013, 2013:66

8. G. E. Vilcu: Canonical foliations on paraquaternionic Cauchy-Riemann submanifolds, Journal of Mathematical Analysis and Applications, 399(2) (2013), 551-558.

9. L. Ornea, M. Parton, P. Piccinni, V. Vuletescu: Spin(9) geometry of the octonionic Hopf fibration, Transformation Groups, 18, No. 3, 845-864 (2013)

10. A.M. Ionescu, V. Slesar, M. Visinescu, G.E. Vilcu, Transversal Killing and twistor spinors associated to the basic Dirac operators, Reviews in Mathematical Physics, vol. 25, Issue 08, September 2013, 21 pages.

11. P. Gauduchon, A. Moroianu, L. Ornea, Compact homogeneous LCK manifolds are Vaisman, Math. Annalen, 361(3) (2015), 1043-1048

12. V. Vuletescu, LCK metrics on Oeljeklaus-Toma manifolds versus Kronecker's theorem, Bull. Math. Soc. Sci. Math. Roum., Nouv. Ser Tome 57(105), No. 2, 225-231, (2014).

13. B.Y. Chen, G.E Vilcu, Geometric classifications of homogeneous production functions, Applied Mathematics and Computation, Volume 225, December 2013, 345-351.

14. V. Slesar, M. Visinescu, G.E. Vilcu Special Killing forms on toric Sasaki-Einstein manifolds Physica Scripta, Volume 89 (2014), Number 12, 125205 (7 pp).

15. L. Ornea, M. Verbitsky, Locally conformally Kahler metrics obtained from pseudoconvex shells, Proc. Amer. Math. Soc. 144 (2016), 325-335.

16. V. Slesar, M. Visinescu, G.E. Vilcu, Toric data, Killing forms and complete integrability of geodesics in Sasaki-Einstein spaces $Y^{p,q}$, Annals of Physics 361 (2015), 548-562.

17. V. Slesar, M. Visinescu, G.E. Vilcu, Hidden symmetries on toric Sasaki-Einstein spaces, EPL (Europhysics Letters) 110(3) (2015), 31001 (6 pp).

18. G.E. Vilcu, On generic submanifolds of manifolds endowed with metric mixed 3-structures, Communications in Contemporary Mathematics (2015) 1550081 (21 pages), DOI: 10.1142/S0219199715500819.

19. L. Ornea, V. Slesar: Basic Morse-Novikov cohomology for foliations, Mathematische Zeitschrift 284 (2016), no. 1-2, 469-489.

20. L. Ornea, M. Verbitsky: LCK rank of locally conformally Kaehler manifolds with potential, J. Geom. Physics. 107 (2016), 92-98.

21. Jae Won Lee, Chul Woo Lee, G.E. Vilcu: Optimal inequalities for the normalized delta-Casorati curvatures of submanifolds in Kenmotsu space forms, to appear in Advances in Geometry.

Papers accepted for publication in BDI indexed journals

1. M. Parton, P. Piccinni, V. Vuletescu: Clifford systems in octonionic geometry, to appear in Rend. Sem. Mat. Torino.

Papers submitted and preprints

1. C. Gherghe, Harmonicity and spectral theory on Sasakian manifolds, preprint, 2013

2. L. Ornea, M. Verbitsky: Hopf surfaces in locally conformally Kaehler manifolds with potential, preprint, 2016.

3. L. Ornea, M. Verbitsky, V. Vuletescu: Weighted Bott-Chern and Dolbeault cohomology for LCK-manifolds with potential, preprint, 2016.

Chapters in books published in foreign editorial houses

1. G.E. Vilcu: Paraquaternionic CR-submanifolds, mixed 3-structures and semi- Riemannian submersions, Chapter 13 in: Geometry of Cauchy-Riemann Submanifolds, Editors: S. Dragomir, M. Hasan Shahid, F. R. Al-Solamy, Springer (2016), 361-390.

Last update: Dec. 06, 2016