Hirzebruch functional equations and complex Krichever genera

TL;DR

This paper characterizes all Krichever genera that are n-multiplicative Hirzebruch genera, showing that only the known two-parametric Todd genus and elliptic functions of level d satisfy this property for all n.

Contribution

It proves that among all Krichever genera, only the two-parametric Todd genus and elliptic functions of level d are n-multiplicative for all n, solving the inverse problem.

Findings

Only these families define n-multiplicative Hirzebruch genera among Krichever genera.

The result characterizes the structure of n-multiplicative genera.

The inverse problem is explicitly solved for all n.

Abstract

It is well known that the two-parametric Todd genus and elliptic functions of level $d$ define $n$-multiplicative Hirzebruch genera, if $d$ divides $n+1$. Both these cases are particular cases of Krichever genera defined by the Baker--Akhiezer functions. In this work the inverse problem is solved. Namely, it is proved that only these families of functions define $n$-multiplicative Hirzebruch genera among all the Krichever genera for all $n$.