Editing Hodge theory (section)

==History==
The field of [[algebraic topology]] was still nascent in the 1920s.  It had not yet developed the notion of [[cohomology]], and the interaction between differential forms and topology was poorly understood.  In 1928, [[Élie Cartan]] published an idea, "''Sur les nombres de Betti des espaces de groupes clos''", in which he suggested—but did not prove—that differential forms and topology should be linked.  Upon reading it, Georges de Rham, then a student, was inspired.  In his 1931 thesis, he proved a result now called [[de Rham's theorem]].  By [[Stokes' theorem]], integration of differential forms along [[singular homology|singular]] chains induces, for any compact smooth manifold ''M'', a bilinear pairing as shown below:
:<math>H_k(M; \mathbf{R}) \times H^k_{\text{dR}}(M; \mathbf{R}) \to \mathbf{R}.</math>
As originally stated,<ref name=glimpse>{{Citation | first1 = Srishti | last1 = Chatterji | last2 = Ojanguren | first2 = Manuel | title = A glimpse of the de Rham era | url = http://sma.epfl.ch/~ojangure/Glimpse.pdf | series = working paper, [[École Polytechnique Fédérale de Lausanne|EPFL]] | year = 2010 | access-date = 2018-10-15 | archive-date = 2023-12-04 | archive-url = https://web.archive.org/web/20231204114020/https://sma.epfl.ch/~ojangure/Glimpse.pdf | url-status = dead }}</ref> de Rham's theorem asserts that this is a [[perfect pairing]], and that therefore each of the terms on the left-hand side are vector space duals of one another.  In contemporary language, de Rham's theorem is more often phrased as the statement that singular cohomology with real coefficients is isomorphic to de Rham cohomology:
:<math>H^k_{\text{sing}}(M; \mathbf{R}) \cong H^k_{\text{dR}}(M; \mathbf{R}).</math>
De Rham's original statement is then a consequence of the fact that over the reals, singular cohomology is the dual of singular homology.

Separately, a 1927 paper of [[Solomon Lefschetz]] used topological methods to reprove theorems of [[Bernhard Riemann|Riemann]].<ref>{{cite journal | last= Lefschetz| first= Solomon | title=Correspondences Between Algebraic Curves | journal= Ann. of Math. (2)| volume= 28| number= 1| year= 1927| pages= 342–354| doi= 10.2307/1968379 | jstor= 1968379 }}</ref> In modern language, if ''ω''<sub>1</sub> and ''ω''<sub>2</sub> are holomorphic differentials on an algebraic curve ''C'', then their [[wedge product]] is necessarily zero because ''C'' has only one complex dimension; consequently, the [[cup product]] of their cohomology classes is zero, and when made explicit, this gave Lefschetz a new proof of the [[Riemann relations]].  Additionally, if ''ω'' is a non-zero holomorphic differential, then <math>\sqrt{-1}\,\omega \wedge \bar\omega</math> is a positive volume form, from which Lefschetz was able to rederive Riemann's inequalities.  In 1929, W. V. D. Hodge learned of Lefschetz's paper.  He immediately observed that similar principles applied to algebraic surfaces.  More precisely, if ''ω'' is a non-zero holomorphic form on an algebraic surface, then <math>\sqrt{-1}\,\omega \wedge \bar\omega</math> is positive, so the cup product of <math>\omega</math> and <math>\bar\omega</math> must be non-zero.  It follows that ''ω'' itself must represent a non-zero cohomology class, so its periods cannot all be zero.  This resolved a question of Severi.<ref>[[Michael Atiyah]], ''William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975'', Biogr. Mem. Fellows R. Soc., 1976, vol. 22, pp. 169–192.</ref>

Hodge felt that these techniques should be applicable to higher dimensional varieties as well.  His colleague Peter Fraser recommended de Rham's thesis to him.  In reading de Rham's thesis, Hodge realized that the real and imaginary parts of a holomorphic 1-form on a [[Riemann surface]] were in some sense dual to each other.  He suspected that there should be a similar duality in higher dimensions; this duality is now known as the [[Hodge star operator]].  He further conjectured that each cohomology class should have a distinguished representative with the property that both it and its dual vanish under the exterior derivative operator; these are now called harmonic forms.  Hodge devoted most of the 1930s to this problem.  His earliest published attempt at a proof appeared in 1933, but he considered it "crude in the extreme".  [[Hermann Weyl]], one of the most brilliant mathematicians of the era, found himself unable to determine whether Hodge's proof was correct or not.  In 1936, Hodge published a new proof. While Hodge considered the new proof much superior, a serious flaw was discovered by Bohnenblust.  Independently, Hermann Weyl and [[Kunihiko Kodaira]] modified Hodge's proof to repair the error.  This established Hodge's sought-for isomorphism between harmonic forms and cohomology classes.

<blockquote>
In retrospect it is clear that the technical difficulties in the existence theorem did not really require any significant new ideas, but merely a careful extension of classical methods. The real novelty, which was Hodge’s major contribution, was in the conception of harmonic integrals and their relevance to algebraic geometry. This triumph of concept over technique is reminiscent of a similar episode in the work of Hodge’s great predecessor Bernhard Riemann.

—[[Michael Atiyah|M. F. Atiyah]], William Vallance Douglas Hodge, 17 June 1903 – 7 July 1975, ''Biographical Memoirs of Fellows of the Royal Society'', vol. 22, 1976, pp. 169–192.
</blockquote>