Abstract
Let X be a smooth projective variety admitting an algebraic vector field V with exactly one zero and a holomorphic C*-action λ so that the condition dλ(t)·V = tpV holds for all t ∈ 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 λ. We discuss several examples, including a construction of the rational Fano 3-folds A′22 and B5 which is due to Konarski [Konarski, J. (1989) in C.M.S. Conference Proceedings, ed. Russell, P. (Am. Math. Soc., Providence, RI), in press].
Keywords: flag variety, cohomology algebra, Poincaré polynomial, principal nilpotent, Fano varieties
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Selected References
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