Extension to Higher Varieties

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 H_y of a v^k+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 U^r 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

$$ J(\underline y ) = \int_{\Gamma } {R(\underline x ,\underline y )} \underline {dx} $$

of the eventual singularities in the x varieties H_y. 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.