Harmonic symmetries for Hermitian manifolds

Wilson, Scott

doi:10.1090/proc/14997

Harmonic symmetries for Hermitian manifolds
HTML articles powered by AMS MathViewer

by Scott O. Wilson PDF

Proc. Amer. Math. Soc. 148 (2020), 3039-3045 Request permission

Abstract:

Complex manifolds with compatible metric have a naturally defined subspace of harmonic differential forms that satisfy Serre, Hodge, and conjugation duality, as well as hard Lefschetz duality. This last property follows from a representation of $\mathfrak {sl}(2,\mathbb {C})$, generalizing the well-known structure on the harmonic forms of compact Kähler manifolds. Some topological implications are deduced.

References

J. Cirici and S.O. Wilson, Dolbeault cohomology for almost complex manifolds, Preprint arxiv:1809.1416 (2018).
J. Cirici and S.O. Wilson, Topological and geometric aspects of almost Kähler manifolds via harmonic theory, Preprint arxiv:1809.1414 (2018).
Jean-Pierre Demailly, Sur l’identité de Bochner-Kodaira-Nakano en géométrie hermitienne, Séminaire d’analyse P. Lelong-P. Dolbeault-H. Skoda, années 1983/1984, Lecture Notes in Math., vol. 1198, Springer, Berlin, 1986, pp. 88–97 (French, with English summary). MR 874763, DOI 10.1007/BFb0077045
Phillip A. Griffiths, The extension problem in complex analysis. II. Embeddings with positive normal bundle, Amer. J. Math. 88 (1966), 366–446. MR 206980, DOI 10.2307/2373200
Dan Popovici, Degeneration at $E_2$ of certain spectral sequences, Internat. J. Math. 27 (2016), no. 14, 1650111, 31. MR 3593673, DOI 10.1142/S0129167X16501111
André Weil, Introduction à l’étude des variétés kählériennes, Publications de l’Institut de Mathématique de l’Université de Nancago, VI. Actualités Sci. Ind. no. 1267, Hermann, Paris, 1958 (French). MR 0111056