group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
A differential form on a Riemannian manifold is called a harmonic form if it is in the kernel of the Laplace operator of in that .
On a compact Kähler manifold the Hodge isomorphism (see there) identifies harmonic differential forms with Dolbeault cohomology classes.
For the moment see at Hodge theory
Georges de Rham, Chapter V of: Differentiable Manifolds – Forms, Currents, Harmonic Forms, Grundlehren 266, Springer (1984) [doi:10.1007/978-3-642-61752-2]
Claire Voisin, section 5 of: Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77 (2002/3)
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