Harmonic form

An exterior differential form $ \alpha $ on a Riemannian manifold $ M $ satisfying the equation $ \Delta \alpha = 0 $, where $ \Delta = d \delta + \delta d $ is the Laplace operator corresponding to the Riemannian metric on $ M $ and $ \delta $ is the adjoint of the exterior differential $ d $. If $ \alpha $ has compact support, its harmonicity is equivalent to $ d \alpha = \delta \alpha = 0 $. The harmonic forms of degree $ p $ on $ M $ form a vector space, $ H ^ {p} ( M) $, over the field $ \mathbf R $. If the Riemannian manifold is compact, $ H ^ {p} ( M) $ is finite-dimensional, being the kernel of the elliptic operator $ \Delta $. Since a harmonic form is closed, de Rham's theorem generates a natural mapping of the space $ H ^ {p} ( M) $ into the real cohomology space $ H ^ {p} ( M, \mathbf R ) $ of degree $ p $ of $ M $. It follows from the Hodge theorem that this mapping is an isomorphism. In particular, harmonic functions, i.e. harmonic forms of degree zero, are constant on a connected compact manifold.

Harmonic forms on a compact Riemannian manifold are invariant with respect to any connected group of isometries of this manifold; for a symmetric space $ M $ the space $ H ^ {p} ( M) $ coincides with the space of $ p $- forms which are invariant with respect to the largest connected group of isometries.

A parallel theory of harmonic forms exists for Hermitian manifolds (cf. Hermitian structure) $ M $. A harmonic form on a Hermitian manifold $ M $ is a complex form lying in the kernel of the Laplace–Beltrami operator $ \square $( cf. Laplace–Beltrami equation). The harmonic forms of type $ ( p, q) $ constitute the space $ H ^ {p,q} ( M) $ over $ \mathbf C $. If $ M $ is compact, $ H ^ {p,q} ( M) $ is finite-dimensional and is naturally isomorphic to the Dolbeault cohomology space. If $ M $ is a Kähler manifold, these two definitions of harmonic forms are really identical, since $ \square = \overline \square \; = \Delta / 2 $. In such a case

$$ H ^ {p, q } ( M) = \ \overline{H}\; {} ^ {q, p } ( M) $$

and

$$ H ^ {k} ( M) \otimes \mathbf C = \ \sum _ {p + q = k } H ^ {p, q } ( M). $$

Let $ \omega $ be the Kähler form on $ M $, let $ L $ be the operator of interior multiplication by $ \omega $, let $ \Lambda $ be the operator adjoint to $ L $, and let $ H _ {0} ^ {p,q} ( M) $ be the space of primitive harmonic forms of type $ ( p, q) $, i.e. forms $ \alpha \in H ^ {p,q} ( M) $ for which $ \Lambda \alpha = 0 $. The following equation is valid for $ p \geq q $ and $ p + q \leq \mathop{\rm dim} _ {\mathbf C} M $:

$$ H ^ {p, q } ( M) = \ \sum _ {s = 0 } ^ { q } L ^ {s} H _ {0} ^ {p - s, q - s } ( M) \cong $$

$$ \cong \sum _ {s = 0 } ^ { q } H _ {0} ^ {p - s, q - s } ( M). $$

For a compact Kähler manifold $ M $ the space $ H ^ {p, 0 } ( M) $ is identical with the space $ \Omega ^ {p} ( M) $ of holomorphic forms (cf. Holomorphic form) of degree $ p $. In particular,

$$ H ^ {1} ( M) \otimes \mathbf C = \ \Omega ^ {1} ( M) + \overline{ {\Omega ^ {1} ( M) }}\; . $$

The study of harmonic functions and forms on Riemann surfaces originates with B. Riemann, whose existence theorems were fully proved at the beginning of the 20th century. The theory of harmonic forms on compact Riemannian manifolds was first presented by W.V.D. Hodge [1].

Various generalizations of the theory of harmonic forms were subsequently given. Let there be given a locally flat (analytic) vector bundle $ E $ on a Riemannian (Hermitian) manifold $ M $, and let there be given a Euclidean (Hermitian) metric on the fibres of $ E $. By suitably generalizing the Laplace (Laplace–Beltrami) operator [4], [8], it is possible to define the spaces $ H ^ {p} ( E) $( $ H ^ {p,q} ( E) $) of harmonic forms with values in $ E $( cf. Differential form). If $ M $ is compact, these spaces are finite-dimensional and isomorphic to the corresponding cohomology spaces of de Rham and Dolbeault, which can in turn be interpreted in terms of sheaf cohomology. In the case of locally flat bundles these cohomology spaces are also closely connected with the cohomology spaces of the group $ \pi _ {1} ( M) $. If $ M $ is not compact, the space of square-integrable harmonic forms is isomorphic to the homology space of the complex of square-integrable forms [2]. If $ M $ is a domain with smooth boundary and compact closure $ \overline{M}\; $ in a Kähler manifold $ \widetilde{M} $, it is also possible to consider the space of harmonic forms of type $ ( p, q) $, with values in an analytic vector bundle $ E $ over $ \widetilde{M} $, smooth in $ M $ and continuous on $ \overline{M}\; $. If $ M $ is strictly pseudo-convex, this space is finite-dimensional and is isomorphic to the Dolbeault cohomology space corresponding to $ E $ over $ M $.

Harmonic forms are a powerful tool in the study of the cohomology of real and complex manifolds and of cohomology spaces of discrete groups. The theory of harmonic forms yields fundamental cohomological properties of compact Kähler manifolds and, in particular, of projective algebraic varieties [1], [4], [5]. Harmonic forms can be used to establish a connection between the curvature of a compact Riemannian manifold and the triviality of some of its cohomology groups [6], [7]. Similar connections have also been obtained in complex analytic geometry [4], [5] and in the theory of discrete transformation groups [8].

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