#### Abstract

By a B-regular variety, we mean a smooth projective variety over C admitting an algebraic action of the upper triangular Borel subgroup B ⊂ SL2(C) such that the unipotent radical in B has a unique fixed point. A result of M. Brion and the first author [4] describes the equivariant cohomology algebra (over C) of a B-regular varietyX as the coordinate ring of a remarkable affine curve in X × P. The main result of this paper uses this fact to classify the B-invariant subvarieties Y of a B-regular varietyX for which the restriction map iY : H ∗(X) → H∗(Y ) is surjective.