Journal of Symplectic Geometry

Volume 17 (2019)

Number 6

On $L_2$-cohomology of almost Hermitian manifolds

Pages: 1773 – 1792

DOI: https://dx.doi.org/10.4310/JSG.2019.v17.n6.a5

Authors

Richard Hind (Department of Mathematics, University of Notre Dame, Indiana, U.S.A.)

Adriano Tomassini (Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Unità di Matematica e Informatica, Università degli studi di Parma, Italy)

Abstract

We prove two results regarding the $L_2$ cohomology of almost complex manifolds. First we show that there exist complete, $d$-bounded almost Kähler manifolds of any complex dimension $n \geq 2$ such that the space of harmonic $1$-forms in $L_2$ has infinite dimension. By contrast a theorem of Gromov [6] states that a complete $d$-bounded Kähler manifold $X$ has no nontrivial harmonic forms of degree different from $n = \operatorname{dim}_{\mathbb{C}} X$. Second let $(X, J, g)$ be a complete almost Hermitian manifold of dimension four. We prove that the reduced $L_2 \: 2^{nd}$-cohomology group decomposes as direct sum of the closure of the invariant and anti-invariant $L_2$-cohomology. This generalizes a decomposition theorem by Drǎghici, Li and Zhang [4] for $4$-dimensional closed almost complex manifolds to the $L_2$-setting.

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1fundingThe first author is partially supported by Simons Foundation grant #317510. The second author is partially supported by the Project PRIN “Varietà reali e complesse: geometria, topologia e analisi armonica”, Project PRIN 2017 “Real and Complex Manifolds: Topology, Geometry and holomorphic dynamics” and by GNSAGA of INdAM.

Received 29 August 2017

Accepted 17 October 2018

Published 17 January 2020