nLab Hodge isomorphism

Idea

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.

Statement

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.

Proposition

There is a canonical homomorphism

$$\mathcal{H}^{p,q}(X) \longrightarrow H^{p,q}(X) \,.$$

If $X$ is compact, then this is an isomorphism, the Hodge isomorphism

References

• 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)