Abstract
The well-known Kähler identities naturally extend to the non-integrable setting. This paper deduces several geometric and topological consequences of these extended identities for compact almost Kähler manifolds. Among these are identities of various Laplacians, generalized Hodge and Serre dualities, a generalized hard Lefschetz duality, and a Lefschetz decomposition, all on the space of d-harmonic forms of pure bidegree. There is also a generalization of Hodge Index Theorem for compact almost Kähler 4-manifolds. In particular, these provide topological bounds on the dimension of the space of d-harmonic forms of pure bidegree, as well as several new obstructions to the existence of a symplectic form compatible with a given almost complex structure.
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Acknowledgements
The authors thank Thomas Holt for pointing out an important sign error that appeared in a preprint of this paper. We would also like to thank the anonymous referee for his suggestions.
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Joana Cirici would like to acknowledge partial support from the AEI/FEDER, UE (MTM2016-76453-C2-2-P) and the Serra Húnter Program. Scott O. Wilson acknowledges support provided by a PSC-CUNY Award, jointly funded by The Professional Staff Congress and The City University of New York.
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Cirici, J., Wilson, S.O. Topological and geometric aspects of almost Kähler manifolds via harmonic theory. Sel. Math. New Ser. 26, 35 (2020). https://doi.org/10.1007/s00029-020-00568-4
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DOI: https://doi.org/10.1007/s00029-020-00568-4
Keywords
- Almost Kähler manifolds
- Kähler identities
- Kähler package
- Hodge decomposition
- Hard Lefschetz
- Harmonic forms
- Almost complex manifolds
- Symplectic manifolds