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

Abstract

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 iY : H*(X) → H*(Y) is surjective.