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On non-pure forms on almost complex manifolds


Authors: Richard Hind, Costantino Medori and Adriano Tomassini
Journal: Proc. Amer. Math. Soc. 142 (2014), 3909-3922
MSC (2010): Primary 32Q60, 53C15, 58A12
DOI: https://doi.org/10.1090/S0002-9939-2014-11578-4
Published electronically: July 22, 2014
MathSciNet review: 3251731
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Abstract: In 2009 T.-J. Li and W. Zhang defined an almost complex structure $ J$ on a manifold $ X$ to be $ \mathcal {C}^\infty $-pure-and-full if the second de Rham cohomology group can be decomposed as a direct sum of the subgroups whose elements are cohomology classes admitting $ J$-invariant and $ J$-anti-invariant representatives. It turns out (see T. Draghici, T.-J. Li and W. Zhang (2010)) that any almost complex structure on a $ 4$-dimensional compact manifold is $ \mathcal {C}^\infty $-pure-and-full. We study the $ J$-invariant and $ J$-anti-invariant cohomology subgroups on almost complex manifolds, possibly non-compact. In particular, we prove an analytic continuation result for anti-invariant forms on almost complex manifolds.


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Additional Information

Richard Hind
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: hind.1@nd.edu

Costantino Medori
Affiliation: Dipartimento di Matematica e Informatica, Università di Parma, Viale Parco Area delle Scienze 53/A, 43124, Parma, Italy
Email: costantino.medori@unipr.it

Adriano Tomassini
Affiliation: Dipartimento di Matematica e Informatica, Università di Parma, Viale Parco Area delle Scienze 53/A, 43124, Parma, Italy
Email: adriano.tomassini@unipr.it

DOI: https://doi.org/10.1090/S0002-9939-2014-11578-4
Keywords: Pure and full almost complex structure, $J$-invariant form, $J$-anti-invariant form
Received by editor(s): August 30, 2011
Received by editor(s) in revised form: October 12, 2012, and December 14, 2012
Published electronically: July 22, 2014
Additional Notes: Partially supported by Fondazione Bruno Kessler-CIRM (Trento) and by GNSAGA of INdAM
Communicated by: Franc Forstneric
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.