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.[5][6][7] They lived at 1 Church Hill Place in the Morningside district.[8]
On his return to Cambridge, he was offered a University Lecturer position in 1933. He became the Lowndean Professor of Astronomy and Geometry at Cambridge, a position he held from 1936 to 1970. He was the first head of DPMMS.
The Theory and Applications of Harmonic Integrals[11] 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 varietyV (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-formα 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.
Hodge also wrote, with Daniel Pedoe, a three-volume work Methods of Algebraic Geometry, on classical algebraic geometry, with much concrete content – illustrating though what Élie Cartan called 'the debauch of indices' in its component notation. According to Atiyah, this was intended to update and replace H. F. Baker's Principles of Geometry.