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 Tr. Mat. Inst. Steklova, 2018, Volume 302, Pages 41–56 (Mi tm3928)

Hirzebruch functional equation: classification of solutions

Elena Yu. Bunkova

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: The Hirzebruch functional equation is $\sum _{i=1}^n\prod _{j\ne i} (1/f(z_j-z_i))=c$ with constant $c$ and initial conditions $f(0)=0$ and $f'(0)=1$. In this paper we find all solutions of the Hirzebruch functional equation for $n\leq 6$ in the class of meromorphic functions and in the class of series. Previously, such results have been known only for $n\leq 4$. The Todd function is the function determining the two-parameter 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 the parameters in $U$ it coincides with the series expansion at zero of $f$. The main results are as follows: (1) Any series solution of the Hirzebruch functional equation for $n=5$ corresponds either to the Todd function or to the elliptic function of level $5$. (2) Any series solution of the Hirzebruch functional equation for $n=6$ corresponds either 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 $\mathbb C\mathrm P^{n-1}$ for $n\leq 6$. A topological application of this study is an effective calculation of the coefficients of elliptic genera of level $N$ for $N=2,…,6$ in terms of solutions of a differential equation with parameters in an irreducible algebraic variety in $\mathbb C^4$.

 Funding Agency Grant Number Russian Science Foundation 14-50-00005 This work is supported by the Russian Science Foundation under grant 14-50-00005.

DOI: https://doi.org/10.1134/S0371968518030032

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 302, 33–47

Bibliographic databases:

UDC: 515.178.2+517.547.58+517.583+517.965

Citation: Elena Yu. Bunkova, “Hirzebruch functional equation: classification of solutions”, Topology and physics, Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday, Tr. Mat. Inst. Steklova, 302, MAIK Nauka/Interperiodica, Moscow, 2018, 41–56; Proc. Steklov Inst. Math., 302 (2018), 33–47

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3928
• https://doi.org/10.1134/S0371968518030032
• http://mi.mathnet.ru/eng/tm/v302/p41

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This publication is cited in the following articles:
1. V. M. Buchstaber, “Cobordisms, manifolds with torus action, and functional equations”, Proc. Steklov Inst. Math., 302 (2018), 48–87
2. Atiyah M. Kouneiher J., “Todd Function as Weak Analytic Function”, Int. J. Geom. Methods Mod. Phys., 16:6 (2019), 1950091
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