The Topological Structure of the Space of Algebraic Varieties

The intention of this note is to announce some recent results concerning the homotopy type of the Chow variety of projective algebraic varieties. We begin by discussing the case of complex projective n-space P n . For fixed integers d > 1 and p, 0 < p < n, let Cp,d(P ) denote the set of effective p-cycles of degree d in P n . This is defined to be the family of all finite sums c = ^2naVa, where for each a, n a is a positive integer and Va C P n is an irreducible subvariety of dimension p, and where degree(c) = J2 <* degree(V^) = d. The set Cp,d(P ) itself carries the structure of a projective algebraic variety. In particular it has a natural topology. The spaces Cp,d(P ), for distinct values of d, are conventionally considered to be mutually disjoint. However, if we fix a "distinguished" p-dimensional linear subspace IQ C P n , we obtain a natural sequence of topological embeddings, c w n c g + i ( P n ) c , given at each level by mapping c to c + JoWe can then consider the union of these spaces C p ( P n ) = l i m C M ( P n ) d