Scope and method of study. In this work we have studied T-invariant rational equivalence in a B-variety X, i.e a smooth projective variety over C with a T-action and a finite set of fixed points of T; T = ( C *)n+1 is the algebraic torus. Our main research goal is to prove that the equivariant k-th Chow group ATk (X) is isomorphic to the ordinary k-th Chow group A k(X), find a computational algorithm for ATk (X), and apply it to some interesting cases.; Findings and conclusions. We have proved that the equivariant k-th Chow group ATk (X) is isomorphic to the ordinary k-th Chow group A k(X). The main theorem in my work gives a necessary and sufficient condition for two T-invariant subvarieties D1, D2 ⊂ X of dimension k to be T-invariantly rationally equivalent using the weights of the characters chi i(t) = ti where t ∈ T and T-invariant subvarieties Z ⊂ X of dimension k + 1. We have investigated the case where the set of fixed components is a finite set of fixed points. As an application of the main theorem we were able to find a basis for the equivariant Chow ring A*T (Hilb P22 ) of the Hilbert scheme Hilb P22 represented by geometric cycles. We were able to find all rational equivalences between this basis and other geometric cycles. Then we calculated all the intersections in the equivariant Chow ring A*T (Hilb P22 ). The application gives evidence that this can be done in general. In other words, for any B-variety X we can find a geometric basis for the equivariant Chow ring A*T (X). Then we can find all rational equivalences between this basis and other geometric cycles, calculate all the intersections in the equivariant Chow ring A*T (X), then forget all about the torus action.
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