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Journal Article
Review Lecture: Periods of Integrals and Topology of Algebraic Varieties
C. T. C. Wall
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 391, No. 1801 (Feb. 8, 1984), pp. 231-254
Published
by: Royal Society
https://www.jstor.org/stable/2397500
Page Count: 24
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Topics: Algebra, Algebraic topology, Mathematical theorems, Mathematical integrals, Curves, Coordinate systems, Vector spaces, Eigenvalues, Triangulation
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Abstract
A major theme of nineteenth century mathematics was the study of integrals of algebraic functions of one variable. This culminated in Riemann's introduction of the surfaces that hear his name and analysis of periods of integrals on cycles on the surface. The creation of a correspondingly satisfactory theory for functions of several variables had to wait on the development of algebraic topology and its application by Lefschetz to algebraic varieties. These results were refined by Hodge's theory of harmonic integrals. A closer analysis of Hodge structures by P. A. Griffiths and P. Deligne in recent years has led to unexpectedly strong restrictions on the topology of the variety and to a diversity of other applications. This advance is closely linked to the study of variation of integrals under deformations, particularly in the neighbourhood of a singular point.
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
© 1984 Royal Society