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On $L_2$-cohomology of almost Hermitian manifolds
- 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
- Subjects :
- Mathematics - Differential Geometry
32Q60, 53C15, 58A12
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1708.06316
- Document Type :
- Working Paper