Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case.
It had not yet developed the notion of cohomology, and the interaction between differential forms and topology was poorly understood.
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,[1] 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.
Separately, a 1927 paper of Solomon Lefschetz used topological methods to reprove theorems of Riemann.
[2] 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.
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 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.
Hermann Weyl, one of the most brilliant mathematicians of the era, found himself unable to determine whether Hodge's proof was correct or not.
Independently, Hermann Weyl and Kunihiko Kodaira modified Hodge's proof to repair the error.
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.
F. Atiyah, William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975, Biographical Memoirs of Fellows of the Royal Society, vol.
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.
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 ΔF = 0 on spacetime, viewed as Minkowski space of dimension 4.
For example, the Hodge theorem implies that the cohomology groups with real coefficients of a closed manifold are finite-dimensional.
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.
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.
By Chow's theorem, complex projective manifolds are automatically algebraic: they are defined by the vanishing of homogeneous polynomial equations on CPN.
Taking wedge products of these harmonic representatives corresponds to the cup product in cohomology, so the cup product with complex coefficients is compatible with the Hodge decomposition: The piece Hp,q(X) of the Hodge decomposition can be identified with a coherent sheaf cohomology group, which depends only on X as a complex manifold (not on the choice of Kähler metric):[8] where Ωp denotes the sheaf of holomorphic p-forms on X.
(If X is projective, Serre's GAGA theorem implies that a holomorphic p-form on all of X is in fact algebraic.)
This is not true for compact complex manifolds in general, as shown by the example of the Hopf surface, which is diffeomorphic to S1 × S3 and hence has b1 = 1.
In this sense, Hodge theory is related to a basic issue in calculus: there is in general no "formula" for the integral of an algebraic function.
The moduli space of all projective K3 surfaces has a countably infinite set of components, each of complex dimension 19.
First, Hodge theory gives restrictions on which topological spaces can have the structure of a smooth complex projective variety.
Second, Hodge theory gives information about the moduli space of smooth complex projective varieties with a given topological type.
The best case is when the Torelli theorem holds, meaning that the variety is determined up to isomorphism by its Hodge structure.
Finally, Hodge theory gives information about the Chow group of algebraic cycles on a given variety.
Namely, the cohomology of any complex algebraic variety has a more general type of decomposition, a mixed Hodge structure.