Abstract
A problem entirely analogous to the extension of a one-dimensional integral beyond an isolated singularity arises for higher dimensional integrals. What is required is an analysis of the behavior of the k-cycles of the sections Hy of a vk+1 around the isolated singularity. (We write now y instead of the earlier z.) As it happens the performance concerned with an irreducible algebraic variety Ur in the space X × Y (“irreducible and “algebraic”) have their obvious meaning. However, the varieties y = const, will only play a minor role. To simplify matters we assume that the generic varieties (dimension p) have no singularities. Our purpose will be precisely to study the effect on certain algebraic integrals
of the eventual singularities in the x varieties Hy. Their singularities are characterized by a locus in the y space known as the Landau variety whose definition is fairly clear and need not be described explicitly. We designate it by the letter L. Historically, L has been introduced by the distinguished Moscow theoretical physicist Liov Davidovich Landau who died a couple of years ago, a victim of an automobile accident.
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© 1975 Springer-Verlag New York Inc.
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Lefschetz, S. (1975). Extension to Higher Varieties. In: Applications of Algebraic Topology. Applied Mathematical Sciences, vol 16. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9367-2_12
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DOI: https://doi.org/10.1007/978-1-4684-9367-2_12
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-90137-4
Online ISBN: 978-1-4684-9367-2
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