Generating Functions in Algebraic Geometry and Sums Over Trees

Abstract

A. Calculate the Betti numbers and Euler characteristics of moduli spaces M 0,n of stable n-pointed curves of genus zero (see e.g. [Ke]), or rather an appropriate generating function for these numbers. B. The same for the space X[n], a natural compactification of the space of n pairwise distinct labelled points on a non-singular compact algebraic variety X constructed for dim X = 1 in [BG] and in general in [FMPh]. (Beilinson and Ginzburg called this space "Resolution of Diagonals", Fulton and MacPherson use the term "Configuration Spaces"). C. Calculate the contribution of multiple coverings in the problem of counting rational curves on Calabi-Yau threefolds (see [AM], [Ko], and more detailed explanations below). All these problems are united by the fact that available algebro-geometric information allows us to represent the corresponding numbers as a sum over trees with markings. M. Kontsevich in [Ko] invoked a general formula of perturbation theory in order to reduce the calculation of the relevant generating functions to the problem of finding the critical value of an appropriate formal potential. We solve problems A and B by applying this formalism in a simpler geometric context than that of [Ko]. Problem C is taken from [Ko]; we were able to directly complete Kontsevich's calculation in this case and obtain a simple closed answer. We will now describe our results (0.3-0.5) and technique (0.6) in some detail. 0.2. General setup. Let Y be an algebraic variety over C, possibly non-smooth and non-compact. Following [FMPh] we denote by P Y (q) the virtual Poincaré polynomial of Y which is uniquely defined by the following properties. a). If Y is smooth and compact, then P Y (q) = j dim H j (Y)q j. (0.1) In particular χ(Y) = P Y (−1). (0.2) b). If X = i X i is a finite union of pairwise disjoint locally closed strata, then P Y (q) = i P Y i (q). (0.3) c). P Y ×Z (q) = P Y (q)P Z (q). It follows that if Y is a fibration over base B with