Conformally symplectic structures and the Lefschetz condition

Mehdi Lejmi; Scott O. Wilson

doi:10.5802/crmath.660

Article de recherche - Géométrie et Topologie

Conformally symplectic structures and the Lefschetz condition
[Les structures conformément symplectiques et la condition de Lefschetz]

Mehdi Lejmi ¹ ; Scott O. Wilson ²

¹ Department of Mathematics, Bronx Community College, City University of New York, 2155 University Av., Bronx, NY 10453, USA.
² Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Blvd., Flushing, NY 11367, USA.

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1491-1495.

Résumé
Abstract

Cette courte note fournit un analogue symplectique au théorème de Vaisman en géométrie complexe. Plus précisement, pour toute variété compacte symplectique satisfaisant la condition de Lefschetz en degré 1, chaque structure localement conformément symplectique est en fait globalement conformément symplectique, quand celle-ci et la forme symplectique sont compatibles avec la même structure presque complexe.

This short note provides a symplectic analogue of Vaisman’s theorem in complex geometry. Namely, for any compact symplectic manifold satisfying the hard Lefschetz condition in degree 1, every locally conformally symplectic structure is in fact globally conformally symplectic, whenever there is a mutually compatible almost complex structure.

Reçu le : 2024-05-02
Révisé le : 2024-06-05
Accepté le : 2024-06-18
Publié le : 2024-11-14

DOI : 10.5802/crmath.660

Classification : 53D05, 53B35
Keywords: Locally conformally symplectic structures, hard Lefschetz condition, almost complex structures
Mot clés : Structures localement conformément symplectiques, condition de Lefschetz, structures presque complexes

Affiliations des auteurs :

Mehdi Lejmi ¹ ; Scott O. Wilson ²

¹ Department of Mathematics, Bronx Community College, City University of New York, 2155 University Av., Bronx, NY 10453, USA.
² Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Blvd., Flushing, NY 11367, USA.

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Conformally symplectic structures and the {Lefschetz} condition},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1491--1495},
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     year = {2024},
     doi = {10.5802/crmath.660},
     language = {en},
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Mehdi Lejmi; Scott O. Wilson. Conformally symplectic structures and the Lefschetz condition. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1491-1495. doi : 10.5802/crmath.660. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.660/

Bibliographie
Cité par

[1] Daniele Angella; Giovanni Bazzoni; Maurizio Parton Structure of locally conformally symplectic Lie algebras and solvmanifolds, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 20 (2020) no. 1, pp. 373-411 | DOI | MR | Zbl

[2] Giovanni Bazzoni Locally conformally symplectic and Kähler geometry, EMS Surv. Math. Sci., Volume 5 (2018) no. 1-2, pp. 129-154 | DOI | MR | Zbl

[3] Chal Benson; Carolyn S. Gordon Kähler and symplectic structures on nilmanifolds, Topology, Volume 27 (1988) no. 4, pp. 513-518 | DOI | MR | Zbl

[4] Chal Benson; Carolyn S. Gordon Kähler structures on compact solvmanifolds, Proc. Am. Math. Soc., Volume 108 (1990) no. 4, pp. 971-980 | DOI | MR | Zbl

[5] Giovanni Bazzoni; Juan Carlos Marrero Locally conformal symplectic nilmanifolds with no locally conformal Kähler metrics, Complex Manifolds, Volume 4 (2017) no. 1, pp. 172-178 | DOI | MR | Zbl

[6] Giovanni Bazzoni; Juan Carlos Marrero On locally conformal symplectic manifolds of the first kind, Bull. Sci. Math., Volume 143 (2018), pp. 1-57 | DOI | MR | Zbl

[7] Joana Cirici; Scott O. Wilson Topological and geometric aspects of almost Kähler manifolds via harmonic theory, Sel. Math., New Ser., Volume 26 (2020) no. 3, 35, 27 pages | DOI | MR | Zbl

[8] L. C. de Andrés; Marisa Fernández; Manuel de León; J. J. Mencía Some six-dimensional compact symplectic and complex solvmanifolds, Rend. Mat. Appl., Volume 12 (1992) no. 1, pp. 59-67 | MR | Zbl

[9] Sorin Dragomir; Liviu Ornea Locally conformal Kähler geometry, Progress in Mathematics, 155, Birkhäuser, 1998, xiv+327 pages | DOI | MR | Zbl

[10] Tedi Drăghici The Kähler cone versus the symplectic cone, Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér., Volume 42(90) (1999) no. 1, pp. 41-49 | MR | Zbl

[11] Yakov Eliashberg; Emmy Murphy Making cobordisms symplectic, J. Am. Math. Soc., Volume 36 (2023) no. 1, pp. 1-29 | DOI | MR | Zbl

[12] Marisa Fernández; Manuel de León; Martín Saralegui A six-dimensional compact symplectic solvmanifold without Kähler structures, Osaka J. Math., Volume 33 (1996) no. 1, pp. 19-35 | MR | Zbl

[13] Tian-Jun Li The space of symplectic structures on closed 4-manifolds, Third International Congress of Chinese Mathematicians. Part 1, 2 (AMS/IP Studies in Advanced Mathematics), Volume 42, American Mathematical Society, 2008, pp. 259-277 | MR | Zbl

[14] Dexie Lin Betti numbers and compatible almost complex structures on the hard Lefschetz condition (2023) (https://arxiv.org/abs/2312.16948)

[15] Iku Nakamura On complex parallelisable manifolds and their small deformations, Proc. Japan Acad., Volume 48 (1972), pp. 447-449 | MR | Zbl

[16] Liviu Ornea; Misha Verbitsky Principles of Locally Conformally Kähler Geometry (2013) (https://arxiv.org/abs/2208.07188)

[17] Nicolleta Tardini; Adriano Tomassini $\bar{\partial}$ -Harmonic forms on $4$ -dimensional almost-Hermitian manifolds (2021)

[18] Qiang Tan; Adriano Tomassini Remarks on some compact symplectic solvmanifolds, Acta Math. Sin., Engl. Ser., Volume 39 (2023) no. 10, pp. 1874-1886 | DOI | MR | Zbl

[19] Adriano Tomassini; Xu Wang Some results on the hard Lefschetz condition, Int. J. Math., Volume 29 (2018) no. 13, 1850095, 30 pages | DOI | MR | Zbl

[20] Izu Vaisman On locally and globally conformal Kähler manifolds, Trans. Am. Math. Soc., Volume 262 (1980) no. 2, pp. 533-542 | DOI | MR | Zbl

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