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Manifolds of solutions for Hirzebruch functional equations. (English. Russian original) Zbl 1331.32014

Proc. Steklov Inst. Math. 290, 125-137 (2015); translation from Tr. Mat. Inst. Steklova 290, 136-148 (2015).
Summary: 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.

MSC:

32Q99 Complex manifolds
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References:

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