Hirzebruch Functional Equation: Classification of Solutions

TL;DR

This paper classifies all solutions to the Hirzebruch functional equation for n ≤ 6 within meromorphic functions and series, identifying solutions related to Todd and elliptic functions, thus completing the classification of certain complex genera.

Contribution

It extends the classification of solutions to the Hirzebruch functional equation from n ≤ 4 to n ≤ 6, identifying all solutions as Todd or elliptic functions of specific levels.

Findings

Solutions for n=5 are Todd or elliptic functions of level 5.

Solutions for n=6 are Todd or elliptic functions of levels 2, 3, or 6.

Complete classification of complex genera with fiber multiplicativity for n ≤ 6.

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 $n \leqslant 6$.