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
Over compact Kähler manifolds $X$, Hodge symmetry is the property that the Dolbeault cohomology groups $H^{p,q}(X)$ are taken into each other under complex conjugation followed by switching the bidegree:
In particular this means that the dimension of the cohomology groups in degree $(p,q)$ – the Hodge number $h^{p,q}$ – coincides with that in bidegree $(q,p)$.
By the Dolbeault theorem this is formulated more generally in terms of abelian sheaf cohomology as saying that
where $\Omega^p$ denotes the abelian sheaf of holomorphic p-forms.
In this form the statement of Hodge symmetry makes sense more generally for complex analytic spaces and schemes over the complex numbers – but it is no longer generally true in these more general cases
Sufficient conditions for Hodge symmetry to hold more generally are discussed for instance in (Joshi 14).
Last revised on June 11, 2014 at 01:56:51. See the history of this page for a list of all contributions to it.