Abstract
Recall that Riemann used closed paths on his surfaces in order to integrate smooth forms of degree 1 along them. In fact, Riemann did not work with arbitrary such forms, but rather with those which may be written as f(z)dz in terms of a local holomorphic coordinate z, the function f(z) being also holomorphic.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
In modern terms, a Riemannian metric on the smooth manifold V is a smooth field of positive definite quadratic forms on the tangent spaces of V at its various points. At that time, the notion of fiber bundle was just emerging (see Chap. 46), and a Riemannian metric was viewed instead in local coordinates (x i) i as an expression ∑ i, j g ij dx i dx j, the multiplication of differentials being commutative, the coefficients g ij being smooth as functions of the variables x i, symmetric, that is, g ij = g ji , and the symmetric matrix (g ij ) i, j being positive definite for all values of the parameters x i.
- 2.
A Kähler metric on a complex manifold is a Riemannian metric g such that the exterior form of degree 2 defined by ω(X, Y ): = g(X, iY ) (where X, Y are any two tangent vectors to the same point) is closed.
- 3.
References
W.P. Barth, K. Hulek, C.A.M. Peters, A. Van de Ven, Compact Complex Surfaces, 2nd Enlarged edn. (Springer, New York, 2004)
A.L. Cauchy, Considérations nouvelles sur les intégrales définies qui s’étendent à tous les points d’une courbe fermée, et sur celles qui sont prises entre des limites imaginaires. C.R. Acad. Sci. Paris 23, 689–702 (1846).
G. de Rham, Quelques souvenirs des années 1925–1950. Cahiers du séminaire d’histoire des mathématiques, vol. 1 (Université Pierre et Marie Curie, Paris, 1980), pp. 19–36. Republished in Œuvres Mathématiques de Georges de Rham (L’Ens. Math., Univ. de Genève, 1981), pp. 651–668
W.V.D. Hodge, The Theory and Applications of Harmonic Integrals (Cambridge University Press, Cambridge, 1941)
W.V.D. Hodge, The topological invariants of algebraic varieties, in Proceedings of the International Congress of Mathematicians (American Mathematical Society, Providence, RI, 1950), pp. 182–192
E. Kähler, Über eine bemerkenswerte Hermitesche Metrik. Abh. Math. Sem. Univ. Hamburg 9 (1), 173–186 (1933)
I.R. Shafarevich, Basic Algebraic Geometry, vol. 2. Schemes and Complex Manifolds 2nd edn. (Springer, New York, 1994)
P. Swinnerton-Dyer, An outline of Hodge theory, in Algebraic Geometry. Oslo 1970 (Proceedings of the Fifth Nordic Summer School in Mathematics) (Wolters-Noordhoff, Groningen, 1972), pp. 277–286
C. Voisin, Hodge Theory and Complex Algebraic Geometry. I., II. Cambridge Studies in Advanced Mathematics, vol. 76 (Cambridge University Press, Cambridge, 2002). Translation from the French by Leila Schneps
A. Weil, Comments on the article “Sur la théorie des formes différentielles attachées à une variété analytique complexe”, in Œuvres scientifiques I (Springer, New York, 1979), pp. 562–564
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Popescu-Pampu, P. (2016). Hodge and the Harmonic Forms. In: What is the Genus?. Lecture Notes in Mathematics(), vol 2162. Springer, Cham. https://doi.org/10.1007/978-3-319-42312-8_42
Download citation
DOI: https://doi.org/10.1007/978-3-319-42312-8_42
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-42311-1
Online ISBN: 978-3-319-42312-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)