The Euler Series of Restricted Chow Varieties

Abstract

Let X be an algebraic projective variety in {\bf P}^n. Denote by {\cal C}_{\lambda} the space of all effective cycles on X whose homology class is \lambda \in H_{2p} (X,{\bf Z}). It is easy to show that {\cal C}_{\lambda} is an algebraic projective variety. Let \chi ({\cal C}_{\lambda} be its Euler characteristic. Define the Euler series of X by E_{p} = \sum_{\lambda\in{C}} \, \chi({\cal C}_{\lambda} \lambda \in \, {\bf Z}[[C]] where {\bf Z}[[C]] is the full algebra over {\bf Z} of the monoid C of all homology classes of effective p-cyles on X. This algebra is the ring of function (with respect the convolution product) over C. Denote by {\bf Z}[C] the ring of functions with finite support on C. We say that an element of {\bf Z}[[C]] is rational if it is the quotient of two elements in {\bf Z}[C]. If a basis for homology is fixed we can associated to any rationa element a rational function and therefore compute the Euler characteristic of {\cal C}_{\lambda}. We prove that E_p is rational for any projective variety endowed with an algebraic torus action in such a way that there are finitely many irreducible invariant subvarieties. If it is smooth we also define the equivariant Euler series and proved it is rational, we relate both series and compute some classical examples. The projective space {\bf P}^n, the blow up of {\bf P}^n at a point, Hirzebruch surfaces, the product of {\bf P}^n with {\bf P}^m.