On the topology of algebraic varieties

Abstract

In this appendix we discuss the basic facts we have used about the cohomology and homology of complex algebraic varieties, including in particular the construction of the class of an algebraic subvariety. Although making constructions like this was one of the main motivating factors in the early development of topology, especially in the work of Poincare and Lefschetz, it is remarkably difficult – nearly a century later – for a student to extract these basic facts from any algebraic topology text. The intuitive way to do this is to appeal to the fact that an algebraic variety can be triangulated, in such a way that its singular locus is a subcomplex; the sum of the top-dimensional simplices, properly oriented, is a cycle whose homology class is the desired class of the subvariety. Making this rigorous and proving the basic properties one needs from this can be done, but that require quite a bit of work. An approach which avoids this difficulty, and has the desirable property of working also on a noncompact ambient space, is to use Borel–Moore homology. This is done in Borel and Haefliger (1961), and in detail in Iversen (1986). That approach, however, is based on sheaf cohomology and sheaf duality. In this appendix, we give an alternative but equivalent formulation which uses only standard facts about singular cohomology. (This is a simplified version of a general construction given in Fulton and MacPherson [1981].