VI.90 William Vallance Douglas Hodge
b. Edinburgh, Scotland, 1903; d. Cambridge, England, 1975
Algebraic geometry; differential geometry; topology
Hodge is famous for his theory of harmonic integrals (or forms), which was described by WEYL [VI.80] as one of the landmarks of twentieth century mathematics.” He was a Scot who spent his early life in Edinburgh but lived for most of his life in Cambridge, where he was Lowndean Professor of Astronomy and Geometry (an archaic title) from 1936 until 1970.
Hodge’s work straddles the area between algebraic geometry, differential geometry, and complex analysis. It can be seen as a natural outgrowth of the theory of RIEMANN SURFACES [III.79] (or algebraic curves) and the work of Lefschetz on the topology of algebraic VARIETIES [IV.4 §7] (of higher dimension). It put algebraic geometry on a modern analytic footing and prepared the ground for the spectacular breakthroughs of the postwar period in the 1950s and 1960s. It also harmonized well with the later interaction with theoretical physics, harking back to the influence of James Clerk Maxwell.
In Riemann surface theory (with one complex dimension), complex structures and real metrics are very closely related and the roots of their relationship can be traced back to the link between the CAUCHY–RIEMANN EQUATIONS [I.3 §5.6] and the LAPLACE OPERATOR [I.3 §5.4]. In higher dimensions this close link disappears and a RIEMANNIAN METRIC [I.3 §6.10] seems alien to complex analysis, but it was Hodge’s great insight to see that real analysis could still play a fruitful role.
Following the formalism of electromagnetic theory as developed by Maxwell, he introduced a generalization of the Laplace operator to exterior DIFFERENTIAL FORMS [III.16] (on any Riemannian manifold) and proved the key theorem that the null space of this operator on r-forms (“harmonic” forms) is naturally isomorphic to the r-dimensional COHOMOLOGY [IV.6 §4] Hr. In other words, a harmonic form is uniquely specified by its periods, and all sets of periods occur.
For complex manifolds, provided the metric is suitably compatible with the complex structure (the KÄHLER CONDITION [III.88 §3], always satisfied by algebraic varieties in projective space), this result can be refined. We get a decomposition of Hr into subspaces Hp,q with p + q = r, with the extreme cases p = r, q = r corresponding to holomorphic or anti-holomorphic forms.
This Hodge decomposition has a rich structure and a wealth of applications. One of the most remarkable is the Hodge signature theorem, which (for an even- dimensional algebraic variety) expresses the signature of the intersection matrix of middle-dimensional cycles in terms of the dimensions of the Hp,q.
Another success of the theory was the characterization of those homology classes of dimension 2n − 2 (on a complex n-manifold) that arise from algebraic subvarieties. He conjectured that a similar characterization would work for all dimensions and proved the easy part. The hard part has resisted all subsequent attempts, and is now one of the million-dollar Millennium Problems of the Clay Institute.
The influence of Hodge’s theory was enormous. First, in algebraic geometry it integrated many classical results into a modern framework and it acted as a launch pad for the subsequent development of modern sheaf theory by Henri Cartan, Serre, and others. Second, it was the first deep result in global differential geometry and paved the way for what became known as “global analysis.” Third, it provided the basis for later developments arising from, or linked to, theoretical physics. These included THE ATIYAH–SINGER INDEX THEOREM [V.2] for elliptic operators, and nonlinear analogues of Hodge theory (the Yang–Mills and the Seiberg–Witten equations), which have played such a key role in the Donaldson theory of four-dimensional manifolds (see DIFFERENTIAL TOPOLOGY [IV.7 §2.5]). More recently, Witten and others have shown how suitable infinite-dimensional versions of Hodge’s theory turn up naturally in QUANTUM FIELD THEORY [IV.17 §2.1.4].
Further Reading
Griffiths, P., and J. Harris. 1978. Principles of Algebraic Geometry. New York: Wiley.
Sir Michael Atiyah