Hodge decomposition
In mathematics, Hodge theory, named after W. V. D. Hodge, uses partial differential equations to study the cohomology groups of a smooth manifold M. The key tool is the Laplacian operator associated to a Riemannian metric on M.
The theory was developed by Hodge in the 1930s as an extension of de Rham cohomology. It has major applications in three settings:
Hodge theory has been particularly powerful in algebraic geometry. For a long time, some of the deepest results of algebraic geometry were only accessible through analytic methods such as Hodge theory. Since the 1980s, some results of Hodge theory for algebraic varieties have also been proved by arithmetic methods, known as p-adic Hodge theory.
The basic version of Hodge theory is about the de Rham complex. Let M be a closed smooth manifold. For a natural number k, let Ωk(M) 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 dk denotes the exterior derivative on Ωk(M). This is a complex in the sense that d2 = 0. De Rham's theorem says that the singular cohomology of M with real coefficients is computed by the de Rham complex:
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