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Hodge Theory and Complex Algebraic Geometry I by Claire Voisin, Leila Schneps

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6

The Case of Kähler Manifolds

When X is a complex manifold equipped with a Hermitian metric, the three Laplacians Δd, Δ, all act on the complex differential forms of X. Here, the Laplacian acts on each space Ap,q(X), which can be seen as the space of (0, q)-forms with values in the vector bundle of holomorphic p-differential forms.

If X is compact, the results of the Δd-harmonic k-forms are in bijection with the cohomology classes of degree ...

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