no. G11
Article de recherche - Géométrie et Topologie
Conformally symplectic structures and the Lefschetz condition
[Les structures conformément symplectiques et la condition de Lefschetz]
Lejmi, Mehdi1 orcid  ; Wilson, Scott O.2 orcid
1Department of Mathematics, Bronx Community College, City University of New York, 2155 University Av., Bronx, NY 10453, USA.
2Department of Mathematics, Queens College, City University of New York, 65-30 Kissena Blvd., Flushing, NY 11367, USA.
Comptes Rendus. Mathématique, Tome 362 (2024) no. G11, pp. 1491-1495

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.

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.

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DOI : 10.5802/crmath.660
Classification : 53D05, 53B35
Keywords: Locally conformally symplectic structures, hard Lefschetz condition, almost complex structures
Mots-clés : Structures localement conformément symplectiques, condition de Lefschetz, structures presque complexes

Lejmi, Mehdi  1   ; Wilson, Scott O.  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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Lejmi, Mehdi; Wilson, Scott O. Conformally symplectic structures and the Lefschetz condition. Comptes Rendus. Mathématique, Tome 362 (2024) no. G11, pp. 1491-1495. doi: 10.5802/crmath.660

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

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

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

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

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

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

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

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

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

[13] Li, Tian-Jun 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] Lin, Dexie Betti numbers and compatible almost complex structures on the hard Lefschetz condition (2023) (https://arxiv.org/abs/2312.16948)

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

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

[17] Tardini, Nicolleta; Tomassini, Adriano ∂ ¯-Harmonic forms on 4-dimensional almost-Hermitian manifolds (2021)

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

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

[20] Vaisman, Izu 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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