- W. V. D. Hodge
William Vallance Douglas Hodge FRS (
17 June 1903- 7 July 1975) was a Scottish mathematician, specifically a geometer.
His discovery of far-reaching topological relations between
algebraic geometryand differential geometry— an area now called Hodge theoryand pertaining more generally to Kähler manifolds — has been a major influence on subsequent work in geometry.
Life and career
He was born in
Edinburgh, attended George Watson's College, and studied at Edinburgh University, graduating in 1923. With help from E. T. Whittakerwhose son J. M. Whittakerwas a college friend, he then took the Cambridge Mathematical Tripos. At Cambridge he fell under the influence of the geometer H. F. Baker.
In 1926 he took up a teaching position at the
University of Bristol, and began work on the interface between the Italian school of algebraic geometry, particularly problems posed by Francesco Severi, and the topological methods of Solomon Lefschetz. This made his reputation, but led to some initial scepticism on the part of Lefschetz. According to Atiyah's memoir, Lefschetz and Hodge in 1931 had a meeting in Max Newman's rooms in Cambridge, to try to resolve issues. In the end Lefschetz was convinced.
In 1930 Hodge was awarded a Research Fellowship at
St. John's College, Cambridge. He spent a year 1931-2 at Princeton University, where Lefschetz was, visiting also Oscar Zariskiat Johns Hopkins University. At this time he was also assimilating de Rham's theorem, and defining the Hodge staroperation. It would allow him to define harmonic forms and so refine the de Rham theory.
On his return to Cambridge, he was offered a University Lecturer position in 1933. He became the
Lowndean Professorof Astronomy and Geometry at Cambridge, a position he held from 1936 to 1970. He was the first head of DPMMS.
He was the Master of
Pembroke College, Cambridgefrom 1958 to 1970, and vice-president of the Royal Societyfrom 1959 to 1965. Amongst other honours, he received the Adams Prizein 1937 and the Copley Medalof the Royal Societyin 1974.
Hodge index theoremwas a result on the intersection numbertheory for curves on an algebraic surface: it determines the signatureof 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 metricof a theory of Laplacians - it applies to an algebraic varietyV (assumed complex, projective and non-singular) because projective spaceitself carries such a metric. In de Rham cohomologyterms, 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
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
This "Hodge decomposition" has become 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.
Further developments by others led in particular to an idea of
mixed Hodge structureon singular varieties, and to deep analogies with étale cohomology.
Hodge conjectureon the 'middle' spaces H"p","p" is still unsolved, in general. It is one of the seven Millennium Prize Problemsset up by the Clay Mathematics Institute.
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 Cartancalled '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".
Michael Atiyah, "William Vallance Douglas Hodge"'. Royal Society biographical memoir, reprinted in Atiyah's Collected Papers Vol.I, pp.231-254
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