Project description

In our recent research, we have showed that some basic but fundamental facts of smooth LCK geometry remain valid in the singular case: we have defined LCK metrics on singular complex spaces, we have proved that the characterization theorem for LCK manifolds is still true for LCK spaces, and also showed that Vaisman’s theorem on the non-existence of genuine LCK metrics on compact Kähler manifolds remains true for locally irreducible compact complex spaces. Therefore, knowing that these fundamental facts of smooth LCK geometry are still true for complex spaces, we are next interested in further developing the theory and studying if the more advanced results in smooth LCK theory have a nice correspondence in the singular case. In this project, we want to study modifications of LCK spaces, images of LCK spaces through holomorphic mappings, special classes of LCK spaces, namely LCK spaces with potential and Vaisman spaces, and, finally, we want to construct some examples of non-trivial LCK spaces.

Expected results

At the end of each phase of the project we will submit for publication our results in high visibility ISI journals. We expect to have finalized and submitted at least 4 articles on the proposed subjects by the end of this project, according to our objectives.

Research team

• Nicolina Istrati

• Alexandra Otiman

• Ovidiu Preda (project coordinator)

• Miron Stanciu

Phase 1 (May 13 - December 31, 2022)

Scientific report for Phase 1 (2022)

Articles:

• N. Istrati & A. Otiman - Bott-Chern cohomology of compact Vaisman manifolds, Trans. Amer. Math. Soc. 376 (2023), pp.3919-3936, arXiv:2206.07312.

• O. Preda & M. Stanciu - Blow-ups and modifications of lcK spaces, Bull. Lond. Math. Soc. 55 (2023), pp.2481-2492, arXiv:2211:10925.

Talks in conferences/seminars:

• A. Otiman - Cohomological and metric properties of OT manifolds - Paris-Sud University (France), May 17, 2022.

• M. Stanciu - Modifications of locally conformally Kähler spaces - Workshop for Young Researchers in Mathematics 11th edition - Bucharest (Romania), May 19-20, 2022.

• N. Istrati - Bott-Chern cohomology of compact Vaisman manifolds, Joint seminar of Complex Algebraic Geometry and Complex Analysis - Münster (Germany), May 27, 2022.

• O. Preda - Locally conformally Kähler structures on complex spaces - Mathematics seminar, Institut for Matematik, Århus Universitet (Denmark), June 7, 2022.

• N. Istrati - Bott-Chern cohomology of compact Vaisman manifolds, Workshop "Young researchers in Kähler and Hermitian geometry" - Montpellier (France), June 2-3, 2022.

• N. Istrati - Bott-Chern cohomology of compact Vaisman manifolds, Conference "Space, Structures, Symmetries", Bari (Italy),  August 28 - September 2, 2022.

Phase 2 (January 1 - December 31, 2023)

Scientific report for Phase 2 (2023)

Articles:

• O. Preda -  lcK spaces. A short survey, Rev. Roumaine Math. Pures Appl. 67 (2023) 1-2, pp.191-198 (link).

• O. Preda & M. Stanciu - Locally conformally Kähler spaces and proper open morphisms, arXiv:2311.14372.

• C. Ciulică, A. Otiman & M. Stanciu - Special non-Kähler metrics on Endo-Pajitnov manifolds arXiv:2403.15618.

Talks in conferences/seminars:

• O. Preda - Blow-ups and modifications of lcK spaces - Workshop for Young Researchers in Mathematics 12th edition - Iasi (Romania), May 18-19, 2023.

• N. Istrati - Vaisman manifolds with vanishing first Chern class - A day of complex geometry - Bucharest (Romania), June 19, 2023.

• O. Preda - Vaisman theorem and local reducibility - A day of complex geometry - Bucharest (Romania), June 19, 2023.

• A. Otiman - Bott-Chern cohomology of compact Vaisman manifolds - 10th Congress of Romanian Mathematicians - Pitesti (Romania), June 30 - July 5, 2023.

• M. Stanciu - Modifications of locally conformally Kähler spaces - 10th Congress of Romanian Mathematicians - Pitesti (Romania), June 30 - July 5, 2023.

• O. Preda - Compact lcK spaces: blow-ups and modifications - 29th Nordic Congress of Mathematicians (with EMS)  - Ålborg (Denmark), July 3-7, 2023.

• O. Preda - Kähler and locally conformally Kähler metrics - Universität Bern, Mathematical Colloquia  - Bern (Switzerland), September 25, 2023.

Research visits:

• O. Preda - Århus University (Denmark), September 10-17, 2023.

• M. Stanciu - Århus University (Denmark), September 10-17, 2023.

Phase 3 (January 1 - May 12, 2024)

Talks in conferences/seminars:

• M. Stanciu - Vaisman's theorem for locally conformally Kähler spaces - Geometry in Timisoara, Timisoara (Romania), March 01-02, 2024.

• O. Preda - Properties of lcK spaces - Workshop Parapluie Angers 2024, Angers (France), April 17-19, 2024.