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Harmonic Integrals Associated with Algebraic Varieties [Received 9 February, 1934. - Read 15 February, 1934.] [In: Proceedings of the London Mathematical Society. Second Series. Volume 39].


London, Hodgson & Son, 1935. Royal 8vo. Volume 39 + 40 of "Proceedings of the London Mathematical Society. Second Series" bound together in a very nice contemporary blue full cloth binding with gilt lettering and gilt ex-libris ("Belford College. Univ. London") to spine. Minor bumping to extremities. Binding tight, and in excellent, very nice, clean, and fresh condition, in- as well as ex-ternally. Small circle-stamp to pasted-down front free end-papers and to title-page ("Bedford College for Women"). Discreet library-markings to upper margin of pasted-down front free end-papers. [Vol. 39:] pp. 249-271. [Entire volume: (Vol. 39:) (4), 546 pp. + (Vol. 40:) (4), 558 pp]. ¶ First publication of Hodge's seminal work on harmonic integrals.
In the article Hodge showed that most of the elementary properties of harmonic functions could be extended to harmonic functional.

"Hodge is famous for his theory of harmonic integrals (or forms), which was described by Weyl as "one of the landmarks of twentieth century mathematics." [...] 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 and the work of Lefschetz on the topology of algebraic varieties. It put the algebraic geometry on a modern analytic footing and prepared the ground for the spectacular breakthroughs of the postwar period of the 1950s and 1960s." (Gowers, The Princeton Companion to Mathematics, 2008).

Hodge himself states in the introduction to the present work: "In two papers I have defined integrals, which I have called harmonic integrals, which are associated with an analytic variety to which a metric is attached, and have established an existence theorem for them. More recently I have applied the theory of these integrals to the Riemannian manifold of an algebraic surface with the topological invariants of the Manifold. There is reason to believe that this method of considering the Abelian integrals attached to an algebraic variety will prove a powerful one, and I have thought it advisable to set out in the following pages an account of the principles on which the method is based."

In 1941 Hodge published the book "The theory and applications of harmonic integrals" which expanded and elaborated the ideas presented in the present article.

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