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 Tr. Mat. Inst. Steklova, 2015, Volume 290, Pages 136–148 (Mi tm3656)

Manifolds of solutions for Hirzebruch functional equations

V. M. Buchstaber, E. Yu. Bunkova

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia

Abstract: For the $n$th Hirzebruch equation we introduce the notion of universal manifold $\mathcal M_n$ of formal solutions. It is shown that the manifold $\mathcal M_n$, where $n>1$, is algebraic and its dimension is not greater than $n+1$. We give a family of polynomials generating the relation ideal in the polynomial ring on $\mathcal M_n$. In the case $n=2$ the generators of this ideal are described. As a corollary we obtain an effective description of the manifold $\mathcal M_2$ and therefore all series determining complex Hirzebruch genera that are fiberwise multiplicative on projectivizations of complex vector bundles. A family of analytic solutions of the second Hirzebruch equation is described in terms of Weierstrass elliptic functions and in terms of Baker–Akhiezer functions of elliptic curves. For this functions the curves differ, yet the series expansions in the vicinity of $0$ coincide.

 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/S0371968515030115

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English version:
Proceedings of the Steklov Institute of Mathematics, 2015, 290:1, 125–137

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Document Type: Article
UDC: 517.9

Citation: V. M. Buchstaber, E. Yu. Bunkova, “Manifolds of solutions for Hirzebruch functional equations”, Modern problems of mathematics, mechanics, and mathematical physics, Collected papers, Tr. Mat. Inst. Steklova, 290, MAIK Nauka/Interperiodica, Moscow, 2015, 136–148; Proc. Steklov Inst. Math., 290:1 (2015), 125–137

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tm3656
• https://doi.org/10.1134/S0371968515030115
• http://mi.mathnet.ru/eng/tm/v290/p136

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This publication is cited in the following articles:
1. V. M. Buchstaber, I. V. Netay, “Hirzebruch Functional Equation and Elliptic Functions of Level $d$”, Funct. Anal. Appl., 49:4 (2015), 239–252
2. E. Yu. Bunkova, “Elliptic function of level $4$”, Proc. Steklov Inst. Math., 294 (2016), 201–214
3. I. V. Netay, “Hirzebruch Functional Equations and Krichever Complex Genera”, Math. Notes, 103:2 (2018), 232–242
4. V. M. Buchstaber, “Cobordisms, manifolds with torus action, and functional equations”, Proc. Steklov Inst. Math., 302 (2018), 48–87
5. Elena Yu. Bunkova, “Hirzebruch functional equation: classification of solutions”, Proc. Steklov Inst. Math., 302 (2018), 33–47
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