group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
The Hodge isomorphism is one of the central statements of Hodge theory, identifying harmonic differential forms as representatives of Dolbeault cohomology.
Together with the the theorem (see at Hodge structure – For a complex analytic spaces – Comparison theorem) which further identifies the corresponding Hodge filtration with that coming from the canonical filtration on the holomorphic de Rham complex, this serves to generalize Hodge theory away from its original home in Kähler geometry to the more general modern theory in complex analytic geometry and more generally algebraic geometry.
For $X$ a Hermitian manifold write $\mathcal{H}^{p,q}(X)$ for the space of $(p,q)$-harmonic differential forms and write $H^{p,q}$ for its Dolbeault cohomology in the bidegree.
There is a canonical homomorphism
If $X$ is compact, then this is an isomorphism, the Hodge isomorphism
Claire Voisin, section 7 of Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77, 2002/3
Zachary Maddock, Dolbeault cohomology (pdf)
Created on June 5, 2014 at 04:06:53. See the history of this page for a list of all contributions to it.