**
Project Executive Summary**

**Research team:**

**Financial and Scientific Reports**

- Post-calculation assessment

- Scientific report

- Post-calculation assessment

- Scientific report

- Post-calculation assessment

- Scientific report

**Research funded by this grant:**

A1. L. Ornea, M. Verbitsky,

A2. C.W. Lee, J.W. Lee, G.E. Vîlcu,

A3. D. Angella, M. Parton, V. Vuletescu,

A4. L. Ornea, V. Slesar,

A5. T. Albu, S. Dăscălescu,

A6. B.-Y. Chen, A.D. Vîlcu, G.-E. Vîlcu,

A7. S.K. Chaubey, G.-E. Vîlcu,

A8. S. Deshmukh, H. Al-Sodais, G.-E. Vîlcu,

A9. L. Ornea, M. Verbitsky,

A10. L. Ornea, M. Verbitsky,

A11. S. Deaconu, V. Vuletescu,

A12. O. Preda, M. Stanciu,

A13. N. Istrati, A. Otiman,

A14. D. Angella, A. Dubickas, A. Otiman, J. Stelzig,

A15. V. Slesar, G.E. Vîlcu,

A16. L. Ornea, M. Verbitsky,

A17. L. Ornea, M. Verbitsky,

A18. L. Ornea, M. Verbitsky,

A19. C. Gherghe, G.-E. Vîlcu,

A20. L. Ornea, M. Verbitsky, V. Vuletescu,

A21. B.-Y. Chen, M.A. Lone, A.-D. Vîlcu, G.-E. Vîlcu,

A22. M. Aquib, M.S. Lone, C. Neacsu, G.-E. Vîlcu,

A23. N. Bin Turki, U.C. De, A.A. Syied, G.-E. Vîlcu,

A24. L. Ornea, M. Verbitsky,

A25. V. Marchidanu,

A26. L. Ornea, M. Verbitsky,

*B. Papers submitted and preprints
*

B1. M.A. Aprodu,

B2. S. Dăscălescu,

B3. M.A. Aprodu,

B4. O. Preda, M. Stanciu,

B5. M.E. Aydin, R. Lopez, G.-E. Vîlcu,

*
C. Chapters in books published in foreign editorial houses*

C1. J.W. Lee, C.W. Lee, B. Sahin, G.-E. Vîlcu,

** 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.