ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 232–242. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © I. V. Netai, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 2, pp. 236–247.
Hirzebruch Functional Equations and Krichever Complex Genera
I. V. Netai
Institute of Information Transmission Problems, Russian Academy of Sciences
(Kharkevich Institute), Moscow, Russia
National Research University Higher School of Economics, Moscow, Russia
Received October 10, 2016; in ﬁnal form, April 4, 2017
Abstract—As is well known, the two-parameter Todd genus and the elliptic functions of level d
deﬁne n-multiplicative Hirzebruch genera if d divides n +1. Both cases are special cases of the
Krichever genera deﬁned by the Baker–Akhiezer function. In the present paper, the inverse problem
is solved. Namely, it is proved that only these properties deﬁne n-multiplicative Hirzebruch genera
among all Krichever genera for all n.
Keywords: Hirzebruch genus, elliptic function, functional equation.
The paper is devoted to studying the solutions of the functional equation
= C (1.1)
with the initial conditions f(0) = 0, f
(0) = 1, which is known as the Hirzebruch functional equation.
The topological interpretation of this equation is well known (for details, see [1, Sec. 4]). From every
formal series f (t)=t + ··· over a coeﬃcient ring R one can construct a Hirzebruch complex genus L
It deﬁnes a ring homomorphism L
is the ring of cobordisms of stably complex
A Hirzebruch genus L
is said to be ﬁberwise multiplicative with respect to bundles of stably
complex manifolds with ﬁber F if for every bundle E → B with ﬁber F we have L
For the case in which the ﬁber F = P
is equal to the complex projective space of dimension n, this
condition is called n-multiplicativity. As was proved in  and [3, Theorem 9.3.6], it is equivalent to
the rigidity property of the genus with respect to the action of the torus on P
. The last condition is
expressed by equation (1.1).
As is well known, the function
deﬁnes an n-multiplicative Hirzebruch genus for every n. It is called the two-parameter Todd genus
and arises naturally in many areas of mathematics, like, for example, enumerative combinatorics and
toric topology. As was proved in , the two-parameter Todd genus is rigid on every almost complex
-manifold. As was shown in , a function f(t) deﬁnes a genus which is rigid on every stably complex
manifold if and only if f(t) is of the form (1.2). It follows from what was said above that this is equivalent
to the multiplicative property in the case of the ﬁber P
. For this case, we shall see below (see Lemma 2)