W. V. D. Hodge - Work

Work

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 solutions of Laplace's equation, one can get unique α. This has the important, immediate consequence of splitting up

Hk(V(C), C)

into subspaces

Hp,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). The dimensions of the subspaces are the Hodge numbers.

This Hodge decomposition has become a fundamental tool. Not only do the dimensions hp,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.

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