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.