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.
Read more about this topic: W. V. D. Hodge
Famous quotes containing the word work:
“A work in progress quickly becomes feral. It reverts to a wild state overnight. It is barely domesticated, a mustang on which you one day fastened a halter, but which now you cannot catch. It is a lion you cage in your study. As the work grows, it gets harder to control; it is a lion growing in strength. You must visit it every day and reassert your mastery over it. If you skip a day, you are, quite rightly, afraid to open the door to its room.”
—Annie Dillard (b. 1945)
“Freedom of enterprise was from the beginning not altogether a blessing. As the liberty to work or to starve, it spelled toil, insecurity, and fear for the vast majority of the population. If the individual were no longer compelled to prove himself on the market, as a free economic subject, the disappearance of this freedom would be one of the greatest achievements of civilization.”
—Herbert Marcuse (1898–1979)
“A poet’s work is to name the unnameable, to point at frauds, to take sides, start arguments, shape the world, and stop it going to sleep.”
—Salman Rushdie (b. 1947)