W. V. D. Hodge

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 Medal of the Royal Society

The Hodge index theorem was a result on the intersection number theory for curves on an algebraic surface: it determines the signature of the corresponding quadratic form. This result was sought by the Italian school of algebraic geometry, 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.