nLab harmonic differential form

Contents

Definition

A differential form $\omega \in \Omega^n(X)$ on a Riemannian manifold $(X,g)$ is called a harmonic form if it is in the kernel of the Laplace operator $\Delta_g$ of $X$ in that $\Delta \omega = (d + d^\dagger)^2 \omega = 0$ .

On a compact Kähler manifold the Hodge isomorphism (see there) identifies harmonic differential forms with Dolbeault cohomology classes.

For the moment see at Hodge theory

Georges de Rham, Chapter V of: Differentiable Manifolds – Forms, Currents, Harmonic Forms, Grundlehren 266, Springer (1984) [doi:10.1007/978-3-642-61752-2]
Claire Voisin, section 5 of: Hodge theory and Complex algebraic geometry I,II, Cambridge Stud. in Adv. Math. 76, 77 (2002/3)
Encyclopedia of Mathematics, Harmonic form

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