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

This article is cited in 1 scientific paper (total in 1 paper)

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:

Document Type: Article
UDC: 515.178.2+517.547.58+517.583+517.965
Received: March 10, 2018

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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\by Elena~Yu.~Bunkova
\paper Hirzebruch functional equation: classification of solutions
\inbook Topology and physics
\bookinfo Collected papers. Dedicated to Academician Sergei Petrovich Novikov on the occasion of his 80th birthday
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\yr 2018
\vol 302
\pages 41--56
\publ MAIK Nauka/Interperiodica
\publaddr Moscow
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\crossref{https://doi.org/10.1134/S0371968518030032}
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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  mathnet  crossref  crossref  isi  elib
  • Труды Математического института им. В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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