Let X be a smooth complex projective variety admitting the following properties:
(1) There exists an algebraic vector field V on X with precisely one zero \(s\in X\), and
(2) there is an algebraic \({\mathbb{C}}^*\)-action \(\lambda: {\mathbb{C}}^*X\to X\) and a positive integer p such that the induced tangent action \(d\lambda\) satisfies the relation \(d\lambda(t)\cdot V=t^ p\cdot V\) for any \(t\in {\mathbb{C}}^*.\)
Examples of such varieties are the projective spaces \({\mathbb{P}}^ n\), the flag varieties G/B defined by a complex semi-simple Lie group G and a Borel subgroup B, the Bott-Samuelson desingularizations of Schubert varieties, and certain Fano threefolds.
In the present paper, the authors prove a product formula for the Poincaré polynomial of varieties satisfying (1) and (2). This formula reads \(P(X,t^{p/2})= \prod^{n}_{i=1}(1-t^{-a_ i+p})/(1-t^{- a_ i}) \); where P denotes the Poincaré polynomial defined by the Betti numbers of X, and \(a_ 1,...,a_ n\) are the weights of \(\lambda\) at the zero s of V, \(n=\dim_{{\mathbb{C}}}X.\)
This explicit product formula for the Poincaré polynomial is then shown to have some amazing consequences: First, in the special case of X being a flag variety G/B of a semi-simple Lie group G, the product formula for the Poincaré polynomial turns out to coincide with the famous Kostant- Macdonald product identity for Lie algebras of maximal tori in B [cf. B. Kostant, Am. J. Math. 81, 973-1033 (1959; Zbl 0099.256)] and I. G. Macdonald, Math. Ann. 199, 161-174 (1972; Zbl 0286.20062)]. - Secondly, it is deduced that if \(\lambda\) and V arise from an algebraic \(SL_ 2({\mathbb{C}})\)-action on X, where X satisfies (1) and (2), then the second Betti number of X is just the multiplicity of the lagest weight of the induced linear \({\mathbb{C}}^*\)-action on the tangent space of X at the zero \(s\in X\). That means, in particular, \(b_ 2(X)\leq \dim_{{\mathbb{C}}}X=n.\)
Besides these general consequences, the authors discuss, at the end of the paper, several concrete examples.