Project Executive Summary

Locally conformally Kähler (LCK) manifolds are 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. They belong to complex differential geometry and can be treated using methods pertaining to complex geometry, Riemannian and conformal geometry, algebraic geometry and topology. The very need of combining all these methods constitutes an intrinsic difficulty of the subject. Our project concerns complex and Riemannian properties of 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.

Budget

Financial and Scientific Reports

2021:- Unique phase
- Post-calculation assessment
- Scientific report

2022:- Unique phase
- Post-calculation assessment
- Scientific report

2023:- Unique phase
- Post-calculation assessment
- Scientific report

Research funded by this grant:

A. Articles published in ISI ranked journals

A1. L. Ornea, M. Verbitsky, Closed orbits of Reeb fields on Sasakian manifolds and elliptic curves on Vaisman manifolds, Mathematische Zeitschrift 299(3), 2287-2296 (2021).

A2. C.W. Lee, J.W. Lee, G.E. Vîlcu, Classification of Casorati ideal Legendrian submanifolds in Sasakian space forms II, Journal of Geometry and Physics 171, 104410 (2022).

A3. D. Angella, M. Parton, V. Vuletescu, On locally conformally Kähler threefolds with algebraic dimension two, International Mathematics Research Notices, rnab36 (2022).

A4. L. Ornea, V. Slesar, Deformations of Vaisman manifolds, Differential Geometry and Its Applications 85, 101940 (2022).

A5. T. Albu, S. Dăscălescu, Free objects and coproducts in categories of posets and lattices, Communications in Algebra 50(7), 3178-3187 (2022).

A6. B.-Y. Chen, A.D. Vîlcu, G.-E. Vîlcu, Classification of graph surfaces induced by weighted-homogeneous functions exhibiting vanishing Gaussian curvature, Mediterranean Journal of Mathematics 19, 162 (2022).

A7. S.K. Chaubey, G.-E. Vîlcu, Gradient Ricci solitons and Fischer–Marsden equation on cosymplectic manifolds, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematica 116, 186 (2022).

A8. S. Deshmukh, H. Al-Sodais, G.-E. Vîlcu, A note on some remarkable differential equations on a Riemannian manifold, Journal of Mathematical Analysis and Applications 519(1), 126778 (2023).

A9. L. Ornea, M. Verbitsky, Non-linear Hopf manifolds are locally conformally Kähler, The Journal of Geometric Analysis (2023), 33, 201 (2023).

A10. L. Ornea, M. Verbitsky, Lee classes on LCK manifolds with potential, Tohoku Mathematical Journal (2022), accepted.

A11. S. Deaconu, V. Vuletescu, On Locally Conformally Kähler metrics on Oeljeklaus-Toma Manifolds, Manuscripta Mathematica (2022), accepted.

A12. O. Preda, M. Stanciu, Vaisman theorem for LCK spaces, Annali della Scuola Normale Superiore di Pisa (2022), accepted.

A13. N. Istrati, A. Otiman, Bott-Chern cohomology of compact Vaisman manifolds, Transactions of the American Mathematical Society (2022), accepted.

A14. D. Angella, A. Dubickas, A. Otiman, J. Stelzig, On metric and cohomological properties of Oeljeklaus-Toma manifolds, Publicacions Matematiques (2022), accepted.

A15. V. Slesar, G.E. Vîlcu, A Vaisman manifolds and transversally Kähler-Einstein metrics, Annali di Matematica Pura ed Applicata (2023), accepted.

A16. L. Ornea, M. Verbitsky, Mall bundles and flat connections on Hopf manifolds, Annales de l'Institut Fourier (2023), accepted.

A17. L. Ornea, M. Verbitsky, Bimeromorphic geometry of LCK manifolds, Proceedings AMS (2023), accepted.

A18. L. Ornea, M. Verbitsky, Compact homogeneous locally conformally Kähler manifolds are Vaisman. A new proof, Riv. Mat. Univ. Parma, Vol. 13 (2022), 439-448.

A19. C. Gherghe, G.-E. Vîlcu, Harmonic maps on locally conformal almost cosymplectic manifolds, Communications in Contemporary Mathematics, accepted; https://doi.org/10.1142/S0219199723500529.

A20. L. Ornea, M. Verbitsky, V. Vuletescu, Do products of compact complex manifolds admit LCK metrics?, Bulletin of the London Mathematical Society, accepted; http://doi.org/10.1112/blms.12962.

A21. B.-Y. Chen, M.A. Lone, A.-D. Vîlcu, G.-E. Vîlcu, Curvature properties of spacelike hypersurfaces in a RW spacetime, Journal of Geometry and Physics, 194 (2023), 105015.

A22. M. Aquib, M.S. Lone, C. Neacsu, G.-E. Vîlcu, On $\delta$-Casorati curvature invariants of Lagrangian submanifolds in quaternionic Kaehler manifolds of constant q-sectional curvature, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematica, 117 (2023), Article number: 107.

A23. N. Bin Turki, U.C. De, A.A. Syied, G.-E. Vîlcu, Investigation of space-times through W_2-curvature tensor in f(R,G) gravity, Journal of Geometry and Physics, 194 (2023), 104987.

A24. L. Ornea, M. Verbitsky, Algebraic cones of LCK manifolds with potential, Journal of Geometry and Physics, 198 (2024), 105103.

A25. V. Marchidanu, Complex structures on the product of two Sasakian manifolds, Journal of Geometry and Physics, 199 (2024), 105134.

A26. L. Ornea, M. Verbitsky, A Calabi-Yau theorem for Vaisman manifolds, To appear in Communications in Analysis and Geometry.

B. Papers submitted and preprints

B1. M.A. Aprodu, Pseudo V-harmonic morphisms, submitted (2022).

B2. S. Dăscălescu, Finiteness conditions for Hopf superalgebras, submitted (2023).

B3. M.A. Aprodu, V-minimal submanifolds, submitted (2023).

B4. O. Preda, M. Stanciu, Locally conformally Kaehler spaces and proper open morphisms, submitted (2023).

B5. M.E. Aydin, R. Lopez, G.-E. Vîlcu, Classification of separable hypersurfaces with constant sectional curvature, submitted (2023).

C. Chapters in books published in foreign editorial houses

C1. J.W. Lee, C.W. Lee, B. Sahin, G.-E. Vîlcu, Chen-Ricci inequalities for Riemannian maps and their applications, in: Differential Geometry and Global Analysis: In Honor of Tadashi Nagano, Editors: B.-Y. Chen et al., Contemporary Mathematics, AMS, vol. 777, 2022, 137--152.

Results obtained:

Within this project, the following important results were obtained:

1. We counted the number of closed elliptic curves on a Vaisman manifold V, proving that their number is either infinite or equal to the sum of all Betti numbers of a Kaehler orbifold obtained as a quasi-regular quotient of V.

2. We gave a new proof for the statement that any homogeneous LCK manifold admits a metric with LCK potential.

3. We extended Vaisman theorem to compact complex spaces with singularities.

4. We constructed a deformation of the Vaisman structure in such a way that the canonical foliation is not affected and the deformation concerns only the transverse orthogonal complement and the transverse Kaehler geometry.

5. We found the set of all possible Lee classes on a manifold admitting an LCK structure with potential.

6. We studied the notion of resonance in Hopf manifolds, and showed that all non-resonant Hopf manifolds are linear; previously, this result was obtained by Kodaira using the Poincare-Dulac theorem.

7. We proved that the Hopf manifolds defined by non-linear holomorphic contractions admit holomorphic embeddings into linear Hopf manifolds, and, moreover they admit LCK metrics.

8. We obtained new cohomological properties of Oeljeklaus-Toma manifolds and showed that they admit new and interesting non-Kaehler, Hermitian metrics.

9. We gave an explicit description of the Bott-Chern cohomology groups of a compact Vaisman manifold in terms of the basic cohomology. We infer that the Bott-Chern numbers and the Dolbeault numbers of a Vaisman manifold determine each other.

10. We completely solved the problem of existence of LCK metrics on Oeljeklaus-Toma manifolds.

11. We obtained a direct proof for the short time existence of the solution for transverse Kaehler-Ricci flow on Vaisman manifolds, but without employing the Molino structure theorem.

12. We established some results on the stability of the identity map on a locally conformal almost cosymplectic manifold.

13. We proved an analogue of the Calabi-Yau theorem for Vaisman structures.

14. We showed that the product of two compact Sasakian manifolds admits a family of complex structures indexed by a complex nonreal parameter, none of whose members admits any compatible locally conformally Kaehler metrics if both Sasakian manifolds are of dimension greater than 1.

15. We proved a stability result for the non-Kaehler geometry of locally conformally Kaehler spaces with singularities.

16. We showed that the product of two LCK manifolds in any of the known classes cannot admit LCK metrics.

17. We showed that LCK manifolds with potential have unique minimal model.

Last update: 12-March-2024