Manifolds of solutions for Hirzebruch functional equations

Abstract

For the nth Hirzebruch equation we introduce the notion of universal manifold Mn of formal solutions. It is shown that the manifold Mn, 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 Mn. In the case n = 2 the generators of this ideal are described. As a corollary we obtain an effective description of the manifold M2 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.