Hirzebruch Functional Equation: Classification of Solutions

Abstract

The Hirzebruch functional equation is \[ \sum_{i = 1}^{n} \prod_{j \ne i} { 1 \over f(z_j - z_i)} = c \] with constant $c$ and initial conditions $f(0)=0, f'(0)=1$. In this paper we find all solutions of the Hirzebruch functional equation for $n \leqslant 6$ in the class of meromorphic functions and in the class of series. Previously, such results were known only for $n \leqslant 4$. The Todd function is the function determining the two-parametric Todd genus (i.e. the $\chi_{a,b}$-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 $U \subset \mathbb{C}^k$ is a series with parameters in the Zariski closure of $U$ in $\mathbb{C}^k$, such that for parameters in $U$ it coincides with the series expansion at zero of $f$. The main results are: Any series solution of the Hirzebruch functional equation for $n = 5$ corresponds to the Todd function or to the elliptic function of level $5$. 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 $\mathbb{C}P^{n-1}$ for $n \leqslant 6$.