King Fahd University of Petroleum and Minerals, KFUPM Box 841, Dhahran 31261, Saudi Arabia.
Proc Natl Acad Sci U S A. 1989 Jun;86(11):3934-7. doi: 10.1073/pnas.86.11.3934.
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].
设 X 为一个光滑射影簇，其上存在一个代数向量场 V，其仅有一个零点且有一个全纯的 C()作用 λ，使得对于所有 t ∈ C()，条件 dlambda(t).V = t(p)V 成立。本文的目的是报告 X 的 Poincaré 多项式的一个乘积公式，当 X 是一个复半单李群的典范簇时，它特别化为经典恒等式 [Formula: see text]。一个惊人的推论是，这样的 X 的第二贝蒂数是线性 C(*)作用在 λ 的汇点处的切空间上的最大权的多重性。我们讨论了几个例子，包括由 Konarski [Konarski, J. (1989) in C.M.S. Conference Proceedings, ed. Russell, P. (Am. Math. Soc., Providence, RI), in press] 构造的有理 Fano 3-流形 A'(22)和 B(5)。
