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Hodge was born in Edinburgh in 1903, the younger son and second of three children of Archibald James Hodge (1869–1938), a searcher of records in the property market and a partner in the firm of Douglas and Company, and his wife, Jane (born 1875), daughter of confectionery business owner William Vallance.
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.
The Kähler identities were first proven by W. V. D. Hodge, appearing in his book on harmonic integrals in 1941. [1] The modern notation of Λ {\displaystyle \Lambda } was introduced by André Weil in the first textbook on Kähler geometry, Introduction à L’Étude des Variétés Kähleriennes.
A mixed Hodge structure on an abelian group consists of a finite decreasing filtration F p on the complex vector space H (the complexification of ), called the Hodge filtration and a finite increasing filtration W i on the rational vector space = (obtained by extending the scalars to rational numbers), called the weight filtration, subject to ...
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge.
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In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by Mochizuki . It bears the name of two mathematicians, Suren Arakelov and W. V. D. Hodge. The main comparison in his theory remains unpublished as ...
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