On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author 4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.