Exponential periods and o-minimality II

TL;DR

This paper completes the comparison of various definitions of exponential periods, establishing their equivalence and connecting naive exponential periods with cohomological and motivic perspectives.

Contribution

It demonstrates that different notions of exponential periods coincide, validating the naive definition as the correct class of exponential periods.

Findings

Naive exponential periods are absolutely convergent.

Naive exponential periods are volumes of definable sets in an o-minimal structure.

Naive exponential periods coincide with periods of exponential Nori motives.

Abstract

This paper is a sequel to "Exponential periods and o-minimality I" that the authors wrote together with Philipp Habegger. We complete the comparison between different definitions of exponential periods, and show that they all lead to the same notion. In the first paper, we show that naive exponential periods are absolutely convergent exponential periods. We also show that naive exponential periods are up to signs volumes of definable sets in the o-minimal structure generated by $\mathbb{Q}$, the real exponential function and ${\sin}|_{[0,1]}$. In this paper, we compare these definitions with cohomological exponential periods and periods of exponential Nori motives. In particular, naive exponential periods are the same as periods of exponential Nori motives, which justifies that the definition of naive exponential periods singles out the correct set of complex numbers to be called exponential periods.