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William Vallance Douglas Hodge ( 17 June 1903 - 7 July 1975) was a Scottish mathematician, specifically a geometer. His discovery of topological relations between algebraic geometry and differential geometry - now called Hodge theory and pertaining more generally to Kähler manifolds - was a major influence on subsequent work. He was born in Edinburgh, and was a professor at Cambridge from 1936 to 1970. Amongst other honours, he received the Copley MedalThe Copley Medal is a scientific award for work in any field of science, the highest award granted by the Royal Society of London. It is also the society's oldest award, the first medal being awarded in 1731. The award was created after a £100 bequest in of the Royal SocietyThe Royal Society of London is claimed to be the oldest learned society still in existence and was founded in 1660. The Royal Irish Academy, founded in 1782, is also closely affiliated with it. The Royal Society of Edinburgh (founded 1783) is a separate S

The Hodge index theorem was a result on the intersection numberIn mathematics, the concept of intersection number arose in algebraic geometry, where two curves intersecting at a point may be considered to 'meet twice' if they are tangent there. In the sense that 'multiple intersections' are limiting cases of n fold i theory for curves on an algebraic surfaceIn mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the complex number field, an algebraic surface is therefore of complex dimension two (as a complex manifold) and so of dimension four as a smooth m: it determines the signatureFor signatures on Wikipedia see Sign your posts on talk pages. In mathematics, see signature (mathematics). A signature is a usually stylized version of someone's name written on documents as a proof of identity, like a seal, but handwritten. Signatures m of the corresponding quadratic formIn mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, the distance between two points in three-dimensional Euclidean space is found by taking the square root of a quadratic form involving six var. This result was sought by the Italian school of algebraic geometryIn relation with the history of mathematics, the Italian school of algebraic geometry refers to the work over half a century or more (flourishing roughly 1885-1935) done internationally in birational geometry, particularly on algebraic surfaces. There wer, but was proved by the topological methods of Lefschetz.

The Theory and Applications of Harmonic Integrals summed up Hodge's development during the 1930s of his general theory. This starts with the existence for any Kähler metric of a theory of Laplacians - it applies to an algebraic variety V (assumed complex, projective and non-singular) because projective space itself carries such a metric. In de Rham cohomology terms, a cohomology class of degree k is represented by a k-form α on V(C). There is no unique representative; but by introducing the idea of harmonic form (Hodge still called them 'integrals'), which are Laplacian solutions, one can get unique α. This has the important, immediate consequence of splitting up H^k(V(C), C) into subspaces H^p,q according to the number p of holomorphic differentials dzⁱ wedged to make up α (the cotangent space being spanned by the dzⁱ and their complex conjugates).

This Hodge decomposition is a fundamental tool. Not only do the dimensions h^p,q refine the Betti numbers, by breaking them into parts with identifiable geometric meaning; but the decomposition itself, as a varying 'flag' in a complex vector space, has a meaning in relation with moduli problems. In broad terms, Hodge theory contributes both to the discrete and the continuous classification of algebraic varieties.

In particular the Hodge conjecture on the 'middle' spaces H^p,p is still unsolved, in general.

Further developments by others led in particular to an idea of mixed Hodge structure on singular varieties, and to deep analogies with étale cohomology .

Hodge also wrote (with Pedoe) a three-volume work on algebraic geometry with much concrete content - but illustrating what Elie Cartan called 'the debauch of indices', in its component notation. In fact a story of Hodge's lecturing style concerned his favouring not only of subscripts and superscripts, but of the letters r and s - which he wrote on a blackboard so as to be indistinguishable.