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].
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机译：令X是一个光滑的射影变种，它接受一个恰好为零且全纯C * -作用λ的代数矢量场V，使得条件dλ（t）·V = t p V满足所有t∈C * 。本注释的目的是报告X的庞加莱多项式的乘积公式，该乘积公式专门针对经典恒等式[公式：参见文本]，当X是半简单复李群的标志变体时。一个令人惊讶的推论是，这样一个X的第二个Betti数是线性C * 作用在X的切点空间在λ汇点处的最大权重的倍数。我们讨论了几个例子，包括由Konarski [Konarski，J.（1989）in C.M. S.会议录，编辑。 Russell，P.（美国数学研究所，罗德岛州普罗维登斯），印刷中。
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