Hodge theory

In mathematics, Hodge theory, named after W. V. D. Hodge, is one aspect of the study of differential forms of a smooth manifold M. More specifically, it works out the consequences for the cohomology groups of M, with real coefficients, of the partial differential equation theory of generalised Laplacian operators associated to a Riemannian metric on M.

It was developed by Hodge in the 1930s as an extension of de Rham cohomology, and has major applications on three levels:

In the initial development, M was taken to be a closed manifold (that is, compact and without boundary). On all three levels, the theory was very influential on subsequent work, being taken up by Kunihiko Kodaira (in Japan and later, partly under the influence of Hermann Weyl, at Princeton) and many others subsequently.

Applications and examples

De Rham cohomology

The original formulation of Hodge theory, due to W. V. D. Hodge, was for the de Rham complex. If M is a compact orientable manifold equipped with a smooth metric g, and Ω^k is the sheaf of smooth differential forms of degree k on M, then the de Rham complex is the sequence of differential operators