A generalization of the Kostant—Macdonald identity

Abstract

Let X be a smooth projective variety admitting an algebraic vector field V with exactly one zero and a holomorphic C(*)-action lambda so that the condition dlambda(t).V = t(p)V holds for all t in C(*). The purpose of this note is to report on a product formula for the Poincaré polynomial of X which specializes to the classical identity [Formula: see text] when X is the flag variety of a semisimple complex Lie group. A surprising corollary is that the second Betti number of such an X is the multiplicity of largest weight of the linear C(*)-action on the tangent space of X at the sink of lambda. We discuss several examples, including a construction of the rational Fano 3-folds A'(22) and B(5) which is due to Konarski [Konarski, J. (1989) in C.M.S. Conference Proceedings, ed. Russell, P. (Am. Math. Soc., Providence, RI), in press].