Manifolds of solutions for Hirzebruch functional equations

V. M. Buchstaber ; E. Yu. Bunkova

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematics, mechanics, and mathematical physics, Tome 290 (2015), pp. 136-148

Résumé

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.