nLab Hodge symmetry

Idea

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:

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

In particular this means that the dimension of the cohomology groups in degree $(p,q)$ – 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

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

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).

References

• Kirti Joshi, Some remarks on Hodge symmetry (arXiv:1402.4176)