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Hodge symmetry

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Complex geometry

Contents

Idea

Over compact Kähler manifolds XX, Hodge symmetry is the property that the Dolbeault cohomology groups H p,q(X)H^{p,q}(X) are taken into each other under complex conjugation followed by switching the bidegree:

H p,q(X)H q,p(X)¯. H^{p,q}(X) \simeq \overline{H^{q,p}(X)} \,.

In particular this means that the dimension of the cohomology groups in degree (p,q)(p,q) – the Hodge number h p,qh^{p,q} – coincides with that in bidegree (q,p)(q,p).

By the Dolbeault theorem this is formulated more generally in terms of abelian sheaf cohomology as saying that

dim H q(X,Ω p)=dim H p(X,Ω q), dim_{\mathbb{C}} H^q(X,\Omega^p) = dim_{\mathbb{C}} H^p(X,\Omega^q) \,,

where Ω p\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).

References

Last revised on June 11, 2014 at 01:56:51. See the history of this page for a list of all contributions to it.