On the equivariant cohomology of subvarieties of a B-regular variety

By a $B$-regular variety, we mean a smooth projective variety over $C$ admitting an algebraic action of the upper triangular Borel subgroup $B \subset SL_2©$suchthattheunipotentradicalin$B$ has a unique fixed point. A result of M. Brion and the first author describes the equivariant cohomology algebra (over $C$) of a $B$-regular variety $X$ as the coordinate ring of a remarkable affine curve in $X\times P^1$. The main result of this paper uses this fact to classify the $B$-invariant subvarieties $Y$ of a $B$-regular variety $X$ for which the restriction map $i_Y:H^{\ast }(X)\to H^{\ast }(Y)$ is surjective.