Hodge theory

In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold M using partial differential equations. The key observation is that, given a Riemannian metric on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic.
The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology. It has major applications in two settings—Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles.
While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in number theory. In arithmetic situations, the tools of p-adic Hodge theory have given alternative proofs of, or analogous results to, classical Hodge theory.

History

The field of algebraic topology was still nascent in the 1920s. It had not yet developed the notion of cohomology, and the interaction between differential forms and topology was poorly understood. In 1928, Élie Cartan published an idea, "Sur les nombres de Betti des espaces de groupes clos", in which he suggested—but did not prove—that differential forms and topology should be linked. Upon reading it, Georges de Rham, then a student, was inspired. In his 1931 thesis, he proved a result now called de Rham's theorem. By Stokes' theorem, integration of differential forms along singular chains induces, for any compact smooth manifold M, a bilinear pairing as shown below:
As originally stated, de Rham's theorem asserts that this is a perfect pairing, and that therefore each of the terms on the left-hand side are vector space duals of one another. In contemporary language, de Rham's theorem is more often phrased as the statement that singular cohomology with real coefficients is isomorphic to de Rham cohomology:
De Rham's original statement is then a consequence of the fact that over the reals, singular cohomology is the dual of singular homology.
Separately, a 1927 paper of Solomon Lefschetz used topological methods to reprove theorems of Riemann. In modern language, if ω₁ and ω₂ are holomorphic differentials on an algebraic curve C, then their wedge product is necessarily zero because C has only one complex dimension; consequently, the cup product of their cohomology classes is zero, and when made explicit, this gave Lefschetz a new proof of the Riemann relations. Additionally, if ω is a non-zero holomorphic differential, then is a positive volume form, from which Lefschetz was able to rederive Riemann's inequalities. In 1929, W. V. D. Hodge learned of Lefschetz's paper. He immediately observed that similar principles applied to algebraic surfaces. More precisely, if ω is a non-zero holomorphic form on an algebraic surface, then is positive, so the cup product of and must be non-zero. It follows that ω itself must represent a non-zero cohomology class, so its periods cannot all be zero. This resolved a question of Severi.
Hodge felt that these techniques should be applicable to higher dimensional varieties as well. His colleague Peter Fraser recommended de Rham's thesis to him. In reading de Rham's thesis, Hodge realized that the real and imaginary parts of a holomorphic 1-form on a Riemann surface were in some sense dual to each other. He suspected that there should be a similar duality in higher dimensions; this duality is now known as the Hodge star operator. He further conjectured that each cohomology class should have a distinguished representative with the property that both it and its dual vanish under the exterior derivative operator; these are now called harmonic forms. Hodge devoted most of the 1930s to this problem. His earliest published attempt at a proof appeared in 1933, but he considered it "crude in the extreme". Hermann Weyl, one of the most brilliant mathematicians of the era, found himself unable to determine whether Hodge's proof was correct or not. In 1936, Hodge published a new proof. While Hodge considered the new proof much superior, a serious flaw was discovered by Bohnenblust. Independently, Hermann Weyl and Kunihiko Kodaira modified Hodge's proof to repair the error. This established Hodge's sought-for isomorphism between harmonic forms and cohomology classes.

In retrospect it is clear that the technical difficulties in the existence theorem did not really require any significant new ideas, but merely a careful extension of classical methods. The real novelty, which was Hodge’s major contribution, was in the conception of harmonic integrals and their relevance to algebraic geometry. This triumph of concept over technique is reminiscent of a similar episode in the work of Hodge’s great predecessor Bernhard Riemann.
—M. F. Atiyah, William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975, Biographical Memoirs of Fellows of the Royal Society, vol. 22, 1976, pp. 169–192.

Hodge theory for real manifolds

De Rham cohomology

The Hodge theory references the de Rham complex. Let M be a smooth manifold. For a non-negative integer k, let Ω^k be the real vector space of smooth differential forms of degree k on M. The de Rham complex is the sequence of differential operators
where d_k denotes the exterior derivative on Ω^k. This is a cochain complex in the sense that . De Rham's theorem says that the singular cohomology of M with real coefficients is computed by the de Rham complex:

Operators in Hodge theory

Choose a Riemannian metric g on M and recall that:
The metric yields an inner product on each fiber by extending the inner product induced by g from each cotangent fiber to its exterior product:. The inner product is then defined as the integral of the pointwise inner product of a given pair of k-forms over M with respect to the volume form associated with g. Explicitly, given some we have
Naturally the above inner product induces a norm, when that norm is finite on some fixed k-form:
then the integrand is a real valued, square integrable function on M, evaluated at a given point via its point-wise norms,
Consider the adjoint operator of d with respect to these inner products:
Then the Laplacian on forms is defined by
This is a second-order linear differential operator, generalizing the Laplacian for functions on Rⁿ. By definition, a form on M is harmonic if its Laplacian is zero:
The Laplacian appeared first in mathematical physics. In particular, Maxwell's equations say that the electromagnetic field in a vacuum, i.e. absent any charges, is represented by a 2-form F such that on spacetime, viewed as Minkowski space of dimension 4.
Every harmonic form α on a closed Riemannian manifold is closed, meaning that. As a result, there is a canonical mapping. The Hodge theorem states that is an isomorphism of vector spaces. In other words, each real cohomology class on M has a unique harmonic representative. Concretely, the harmonic representative is the unique closed form of minimum L² norm that represents a given cohomology class. The Hodge theorem was proved using the theory of elliptic partial differential equations, with Hodge's initial arguments completed by Kodaira and others in the 1940s.
For example, the Hodge theorem implies that the cohomology groups with real coefficients of a closed manifold are finite-dimensional. Indeed, the operators Δ are elliptic, and the kernel of an elliptic operator on a closed manifold is always a finite-dimensional vector space. Another consequence of the Hodge theorem is that a Riemannian metric on a closed manifold M determines a real-valued inner product on the integral cohomology of M modulo torsion. It follows, for example, that the image of the isometry group of M in the general linear group is finite.
A variant of the Hodge theorem is the Hodge decomposition. This says that there is a unique decomposition of any differential form ω on a closed Riemannian manifold as a sum of three parts in the form
in which γ is harmonic:. In terms of the L² metric on differential forms, this gives an orthogonal direct sum decomposition:
The Hodge decomposition is a generalization of the Helmholtz decomposition for the de Rham complex.

Hodge theory of elliptic complexes

and Bott defined elliptic complexes as a generalization of the de Rham complex. The Hodge theorem extends to this setting, as follows. Let be vector bundles, equipped with metrics, on a closed smooth manifold M with a volume form dV. Suppose that
are linear differential operators acting on C^∞ sections of these vector bundles, and that the induced sequence
is an elliptic complex. Introduce the direct sums:
and let L be the adjoint of L. Define the elliptic operator. As in the de Rham case, this yields the vector space of harmonic sections
Let be the orthogonal projection, and let G be the Green's operator for Δ. The Hodge theorem then asserts the following:

H and G are well-defined.
Id = H + ΔG = H + GΔ
LG = GL, L''G'' = GL
The cohomology of the complex is canonically isomorphic to the space of harmonic sections,, in the sense that each cohomology class has a unique harmonic representative.

There is also a Hodge decomposition in this situation, generalizing the statement above for the de Rham complex.