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

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5

Harmonic Forms and Cohomology

In the preceding chapter, we showed that the de Rham cohomology groups of a differentiable manifold X were topological invariants. We will now show that if we also have a Riemannian structure on the manifold X (which we assume compact), it is possible to exhibit representatives, which are particular closed differential forms, for the de Rham cohomology classes. These differential forms, which are called harmonic forms, are not only closed, but satisfy another first order differential equation: they are coclosed, i.e. annihilated by the formal adjoint d* of the operator d.

Since the manifold X is compact, the metric on X provides a metric (·, ·)L2, the L2 metric, on the spaces Ak(X) of differential forms. ...

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