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 i _Y : H ^*(X) → H ^*(Y) is surjective.