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On $L_2$-cohomology of almost Hermitian manifolds

Authors :
Hind, Richard
Tomassini, Adriano
Publication Year :
2017

Abstract

Let $(X,J,\omega,g)$ be a complete $n$-dimensional K\"ahler manifold. A Theorem by Gromov \cite{G} states that the if the K\"ahler form is $d$-bounded, then the space of harmonic $L_2$ forms of degree $k$ is trivial, unless $k=\frac{n}{2}$. Starting with a contact manifold $(M,\alpha)$ we show that the same conclusion does not hold in the category of almost K\"ahler manifolds. 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 Dr\v{a}ghici, Li and Zhang \cite{DLZ} for $4$-dimensional closed almost complex manifolds to the $L_2$-setting.<br />Comment: 13 pages

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.1708.06316
Document Type :
Working Paper