On T-Equivariant Rational Equivalence of T-Equivariant Cycles

Abstract

Let m be the m-dimensional projective space over the field of complex numbers acted on by the algebraic torus T m 1 . Let X be an equivariantly embedded variety in m via the map , i.e., is an embedding and t . x X whenever t T and x X . We begin a study of T-equivariant rational equivalence of T-equivariant cycles on a B-variety X, namely a non-singular projective variety with an action of a torus T m 1 with many fixed points. First, we compute the T-equivariant Picard group of such a variety with the help of fixed point and their associated characters. This result is then applied to study more generally Tequivariant rational equivalence of T-equivariant cycles.