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Concept# Hodge theory

Summary

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.
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 a note entitled 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 immediately struck by inspiration. In his 1931 thesis, he proved a spectacular 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 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.

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De Rham cohomology

In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold.

Coherent sheaf cohomology

In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaf cohomology is a technique for producing functions with specified properties. Many geometric questions can be formulated as questions about the existence of sections of line bundles or of more general coherent sheaves; such sections can be viewed as generalized functions. Cohomology provides computable tools for producing sections, or explaining why they do not exist. It also provides invariants to distinguish one algebraic variety from another.

Dolbeault cohomology

In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q). Let Ωp,q be the vector bundle of complex differential forms of degree (p,q).

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The goal of this course is to help students learn the basic theory of complex manifolds and Hodge theory.

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