geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Serre duality in complex analytic geometry is the duality induced by the Hodge star operator on the Dolbeault complex. This generalizes to suitable non-singular projective algebraic varieties over other base rings.
Let $X$ be a Hermitian manifold of complex dimension $dim_{\mathbb{C}}(\Sigma) = n$. Its Riemannian metric induces a Hodge star operator which acts on the pieces in the Dolbeault complex as
(see at Hodge star operator – On a Kähler manifold).
Moreover, complex conjugation gives $\mathbb{C}$-antilinear functions
Write
for the composite antilinear function.
e.g. (Huybrechts 04, def. 4.1.6)
By the basic properties of the Hodge star it follows that restricted to $\Omega^{p,q}(X)$
For $X$ a compact Hermitian manifold, define a bilinear form
as the integration of differential forms
of the wedge product of $\alpha$ with the image of $\beta$ under the complex conjugated Hodge star operator of def. .
(Serre duality)
The pairing of def. induces a non-degenerate sesquilinear (i.e. hermitian) form on Dolbeault cohomology
e.g. (Huybrechts 04, prop 4.1.15)
For $\Sigma$ a compact Kähler manifold the Hodge theorem gives an isomorphism
between the ordinary cohomology of the underlying topological space with coefficients in the complex numbers, and the direct sum of all the Dolbeault cohomology groups in the same total degree.
Therefore for $\Sigma$ of complex dimension $dim_{\mathbb{C}}(\Sigma)= n$ then Serre duality in the form of prop. induces an isomorphism in ordinary cohomology of the form
The isomorphism in remark coincides with Poincaré duality.
Discussion in complex analytic geometry (Hermitian manifolds) includes
Daniel Huybrechts around prop. 4.1.15 of Complex geometry - an introduction. Springer (2004). Universitext. 309 pages. (pdf)
R.O. Wells, Differential Analysis on Compact Manifolds, Second Edition, Springer, 1980. 14
and review with emphasis on the case of Kähler manifolds includes