Hirzebruch Functional Equation: Classification of Solutions

Abstract

The Hirzebruch functional equation iswith constant c and initial conditions f (0) = 0, f ′ (0) = 1. In this paper we find all solutions of the Hirzebruch functional equation for n 6 in the class of meromorphic functions and in the class of series. Previously, such results were known only for n 4. The Todd function is the function determining the two-parametric Todd genus (i.e. the χ a,b genus). It gives a solution to the Hirzebruch functional equation for any n. The elliptic function of level N is the function determining the elliptic genus of level N . It gives a solution to the Hirzebruch functional equation for n divisible by N .A series corresponding to a meromorphic function f with parameters in U ⊂ C k is a series with parameters in the Zariski closure of U in C k , such that for parameters in U it coincides with the series expansion at zero of f . The main results are:Theorem. 10.2. Any series solution of the Hirzebruch functional equation for n = 5 corresponds to the Todd function or to the elliptic function of level 5.Theorem. 11.3. Any series solution of the Hirzebruch functional equation for n = 6 corresponds to the Todd function or to the elliptic function of level 2, 3 or 6.This gives a complete classification of complex genera that are fiber multiplicative with respect to CP n−1 for n 6.