Editing Hodge theory (section)

{{Short description|Mathematical manifold theory}}
In [[mathematics]], '''Hodge theory''', named after [[William Vallance Douglas Hodge|W. V. D. Hodge]], is a method for studying the [[cohomology group]]s of a [[smooth manifold]] ''M'' using [[partial differential equation]]s.  The key observation is that, given a [[Riemannian metric]] on ''M'', every cohomology class has a [[representative (mathematics)|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 manifold]]s and [[Kähler manifold]]s.  Hodge's primary motivation, the study of complex [[projective variety|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 cycle]]s.

While Hodge theory is intrinsically dependent upon the real and [[complex number]]s, it can be applied to questions in [[number theory]].  In arithmetic situations, the tools of [[p-adic Hodge theory|''p''-adic Hodge theory]] have given alternative proofs of, or analogous results to, classical Hodge theory.