W. V. D. Hodge

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 geometry and differential geometry — an area now called Hodge theory and 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. Whittaker whose son J. M. Whittaker was 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 Zariski at Johns Hopkins University. At this time he was also assimilating de Rham's theorem, and defining the Hodge star operation. 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 Professor of 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, Cambridge from 1958 to 1970, and vice-president of the Royal Society from 1959 to 1965. Amongst other honours, he received the Adams Prize in 1937 and the Copley Medal of the Royal Society in 1974.


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


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 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 structure on singular varieties, and to deep analogies with étale cohomology.

Hodge conjecture

The Hodge conjecture on the 'middle' spaces H"p","p" is still unsolved, in general. It is one of the seven Millennium Prize Problems set 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 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".

ee also

*Hodge dual


* Michael Atiyah, "William Vallance Douglas Hodge"'. Royal Society biographical memoir, reprinted in Atiyah's Collected Papers Vol.I, pp.231-254

External links

*MathGenealogy |id=18583

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Hodge — ist der Familienname folgender Personen: Aldis Hodge (* 1986), US amerikanischer Schauspieler Brian Hodge (* 1960), US amerikanischer Horrorschriftsteller Charles Hodge (1797–1878), US amerikanischer Theologe Charlie Hodge (* 1933), kanadischer… …   Deutsch Wikipedia

  • Hodge — may refer to:People with the surname Hodge: *Al Hodge (rock musician) (1951–2006), guitarist and songwriter *Al Hodge (1912–1979), American actor *Archibald Alexander Hodge (1823–1886), American Presbyterian leader *Bob Hodge (born 1955),… …   Wikipedia

  • Hodge (cat) — Hodge was one of Samuel Johnson s cats, immortalized in a characteristically whimsical passage in James Boswell s Life of Johnson . Although there is little known about Hodge, such as his life, his death, or any other information, what is known… …   Wikipedia

  • Hodge (Luisiana) — Hodge Villa de los Estados Unidos …   Wikipedia Español

  • Hodge 301 — is a star cluster in the Tarantula Nebula, visible from Earth s Southern Hemisphere. The cluster and nebula lie about 168,000 light years away, in one of our Milky Way Galaxy s orbiting satellite galaxies, the Large Magellanic Cloud.Hodge 301,… …   Wikipedia

  • Hodge-podge — or Hodgepodge may refer to:* Hodge Podge (character), a fictitious character from the comic strip Bloom County * Hodge Podge (soup), a mutton soup * Hodgepodge , a Neal Morse albumee also* Hotch pot Hodge podge a bunch of left overs thrown… …   Wikipedia

  • Hodge-Podge (character) — Hodge Podge the rabbit is a fictitious character from Berke Breathed s comic strip Bloom County. His best friends are Portnoy and Cutter John. Hodge is extremely conservative and fanatical about most things, though often ignorant and naive about… …   Wikipedia

  • Hodge theorem — may refer to:*Hodge theory *Hodge index theorem …   Wikipedia

  • HODGE Hugh Lenox — (1796 1973) (retrato) [véase http://www.iqb.es/diccio/h/ho.htm#hodge]: ginecólogo norteamericano inventor del pesario de Smith Hodge en forma de anillo ovalado curvo en uno de sus extremos propio para las retrodesviaciones Diccionario ilustrado… …   Diccionario médico

  • Hodge-Pessar — [họds̶c̶h̶...; nach dem amer. Gynäkologen H.Hodge PessarL. Hodge, 1796 1873]: gebogener Ring, der vor allem bei älteren Frauen zur Behebung einer Retroflexion der Gebärmutter in die Scheide eingelegt wird …   Das Wörterbuch medizinischer Fachausdrücke

  • Hodge — Hodge, LA U.S. village in Louisiana Population (2000): 492 Housing Units (2000): 273 Land area (2000): 0.958023 sq. miles (2.481267 sq. km) Water area (2000): 0.409527 sq. miles (1.060670 sq. km) Total area (2000): 1.367550 sq. miles (3.541937 sq …   StarDict's U.S. Gazetteer Places

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.