LOCALLY CONFORMALLY KÄHLER GEOMETRY AND RELATED STRUCTURES
PROGRAM PNIIIDPCE201130118 CONTRACT NR.
2805/05.10.2011
Financed by UEFISCDI  Romanian Executive Agency for Higher Education, Research, Development
and Innovation
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 2form associated to the metric,
Ω(X, Y)=g(X, JY)
obeys a relation of the type
dΩ=θ ∧ Ω
where θ is a closed 1form (called the Lee form).
Locally conformally Kähler manifolds have been intensively studied in the last three decades. Today, 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 nonKähler 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 socalled Vaisman manifolds (and their small deformations) and the OeljeklausToma 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 OeljeklausToma manifolds. These nonKähler 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 (OrneaVerbitsky) that these manifolds have, like the Vaisman ones, a 2dimensional 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 nonVaisman 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:
Blowups of Locally Conformally Kahler Manifolds,
International Mathematics Research Notices, 12 (2013), 28092821.
2. G. E. Vilcu:
Ruled CRsubmanifolds of locally conformal Kahler manifolds,
Journal of Geometry and Physics, 62 (6) (2012), 13661372.
3. M Visinescu, G. Vilcu:
Hidden symmetries of Euclideanised KerrNUT(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), 4853
5. L. Ornea, M. Parton, V. Vuletescu:
Holomorphic
submersions of locally conformally Kahler manifolds,
Annali di Matematica Pura ed Applicata,
193 (5), (2104),13451351
6. G. E. Vilcu:
Mixed
paraquaternionic 3submersions, Indagationes Mathematicae,
24(2) (2103), 474488
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 CauchyRiemann
submanifolds,
Journal of Mathematical Analysis and Applications, 399(2) (2013),
551558.
9. L. Ornea, M. Parton, P. Piccinni, V. Vuletescu:
Spin(9) geometry of the octonionic Hopf fibration,
Transformation Groups, 18, No. 3, 845864 (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), 10431048
12. V. Vuletescu,
LCK metrics on OeljeklausToma manifolds versus Kronecker's theorem,
Bull. Math. Soc. Sci. Math. Roum., Nouv. Ser Tome 57(105), No. 2, 225231, (2014).
13. B.Y. Chen, G.E Vilcu,
Geometric classifications of homogeneous production functions,
Applied Mathematics and Computation, Volume 225, December 2013, 345351.
14. V. Slesar, M. Visinescu, G.E. Vilcu
Special Killing forms on toric SasakiEinstein 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), 325335.
16. V. Slesar, M. Visinescu, G.E. Vilcu,
Toric data, Killing forms and complete integrability of geodesics in SasakiEinstein spaces $Y^{p,q}$,
Annals of Physics 361 (2015), 548562.
17. V. Slesar, M. Visinescu, G.E. Vilcu,
Hidden symmetries on toric SasakiEinstein spaces,
EPL (Europhysics Letters) 110(3) (2015), 31001 (6 pp).
18. G.E. Vilcu,
On generic submanifolds of manifolds endowed with metric mixed 3structures,
Communications in Contemporary Mathematics (2015) 1550081 (21 pages), DOI: 10.1142/S0219199715500819.
19. L. Ornea, V. Slesar:
Basic MorseNovikov cohomology for foliations,
Mathematische Zeitschrift 284 (2016), no. 12, 469489.
20. L. Ornea, M. Verbitsky:
LCK rank of locally conformally Kaehler manifolds with
potential,
J. Geom. Physics. 107 (2016), 9298.
21. Jae Won Lee, Chul Woo Lee, G.E. Vilcu:
Optimal inequalities for the normalized deltaCasorati 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 BottChern and Dolbeault
cohomology for LCKmanifolds with potential, preprint, 2016.
Chapters in books published in foreign editorial houses
1. G.E. Vilcu:
Paraquaternionic CRsubmanifolds, mixed 3structures and semi
Riemannian submersions, Chapter 13 in: Geometry of
CauchyRiemann Submanifolds, Editors: S. Dragomir, M. Hasan Shahid, F. R. AlSolamy, Springer (2016), 361390.
Last update: Dec. 06, 2016