Abstract
AbstractWe consider several differential operators on compact almost-complex, almost-Hermitian and almost-Kähler manifolds. We discuss Hodge Theory for these operators and a possible cohomological interpretation. We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies.
Highlights
On a complex manifold X the exterior derivative d decomposes as the sum of two other cohomological di erential operators, namely d = ∂ + ∂ ̄ satisfying ∂ =, ∂ ̄ = and ∂∂ ̄ + ∂ ̄∂ =
We compare the associated spaces of harmonic forms and cohomologies with the classical de Rham, Dolbeault, Bott-Chern and Aeppli cohomologies
Once a Hermitian metric on X is xed one can associate to ∂ ̄ a natural elliptic di erential operator, the Dolbeault Laplacian; if X is compact the kernel of this operator has a cohomological interpretation, i.e., it is isomorphic to the Dolbeault cohomology of X
Summary
If we do not assume the integrability of the almost-complex structure, i.e., (X, J) is an almost-complex manifold, the ∂ ̄ operator is still well-de ned but it has no more a cohomological meaning. In this paper we are interested in studying the properties of such operators, their harmonic forms and possibly their cohomological meaning on compact manifolds endowed with a non-integrable almost-complex structure. The considered parametrized cohomology groups do not generalize (except for the almost-Kähler case) the classical Dolbeault, Bott-Chern and Aeppli cohomology groups of complex manifolds. Let (X, J) be an almost-complex manifold and consider a linear combination of the di erential operators ∂ , ∂ ̄ , μ , μ, Da,b,c,e := a ∂ ̄ + b ∂ + c μ + e μ , with a, b, c, e ∈ C \ { }.
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