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Integrals of the Second Kind on an Algebraic Variety

1955; Princeton University; Volume: 62; Issue: 1 Linguagem: Inglês

10.2307/2007100

1939-8980

W. V. D. Hodge, Michael Atiyah,

History and Theory of Mathematics

The classical theory of rational functions on a curve, and their integrals, was generalised by Picard to give a theory of functions and their integrals on an algebraic surface, in an extensive series of memoirs, later included in Picard and Simart [141. A large part of this book is devoted to the theory of double integrals of the second kind and their relation with the simple integrals of the third kind. Later, Lefschetz took up this subject again. His great contribution was to make a thorough study of the topology of an algebraic surface, and then, by using this, to clarify the nature of the possible singularities of integrals and so obtain a simplified version of the theory of integrals of the second kind. An account of his work will be found in [12]. Subsequently, using the same topological methods, he carried the work over to the study of integrals on a variety of three dimensions [111, and he indicated, in general terms, how his methods could be extended to varieties of m dimensions, and the nature of the results to be expected. This extension has never been carried through explicitly, and, although it is generally felt that the main obstacle to this is organizational, it is clear that a number of grave technical difficulties remain to be overcome. The development of the theory of stacks or.faisceaux1 by Leray [13], and its adaptation for use in the theory of functions of several complex variables by Cartan and others [11, has provided us with a new tool for dealing with integrals of the second and the purpose of this paper is to show how it leads to a complete theory of these integrals. This is one of the problems mentioned in Hirzebruch [4]. A summary of our results has appeared elsewhere (Comptes Rendus, 239 (1954)). As might be expected, one of the major difficulties has been to frame a definition of integrals of the second or rather of the differential forms which are their integrands, which is suitable for stack-theoretic methods. A preliminary study of certain special cases suggested such a definition, and this soon proved to be the natural definition for use in stack theory. But it is not obvious that this definition coincides with that of Picard and Lefschetz in all cases. While the theory as developed by the method of stacks is of some interest in its own right, we have felt that it is desirable to devote a considerable amount of space to reconciling our definition with that of Picard and Lefschetz, both in order to justify the use of the name integral of the second kind, and to show that our work is indeed a contribution to one of the classical problems of algebraic geometry.