Exponential periods and o-minimality

TL;DR

The paper studies exponential periods, numbers arising as integrals of $e^{-f}\omega$ with algebraic data, and establishes a comprehensive equivalence between naive and cohomological notions of these periods over subfields $k$ with $k$ algebraic over $k_0=k\cap\mathbb{R}$. It proves that for such $k$, naive exponential periods coincide with cohomological exponential periods and with periods of effective exponential motives, while also showing that the real and imaginary parts of these numbers are volumes of compact sets definable in the o-minimal structure $\mathbb{R}_{\sin,\exp}$ over $k_0$. The main technical machinery combines rapid decay homology, twisted de Rham cohomology, real oriented blow-ups, and semi-algebraic/definable triangulations, organized via hypercovers to handle general (possibly singular) varieties and relative data. A key outcome is a robust period isomorphism in the exponential setting and a bridge between period theory and tame geometry, suggesting new directions for characterizing exponential periods through definable volumes and motivic frameworks. The results illuminate deep connections between arithmetic geometry, o-minimality, and the theory of exponential motives, with potential applications to explicit computations and to broader questions about the nature of transcendental numbers arising as period values.

Abstract

Let $α\in \mathbb{C}$ be an exponential period. We show that the real and imaginary part of $α$ are up to signs volumes of sets definable in the o-minimal structure generated by $\mathbb{Q}$, the real exponential function and ${\sin}|_{[0,1]}$. This is a weaker analogue of the precise characterisation of ordinary periods as numbers whose real and imaginary part are up to signs volumes of $\mathbb{Q}$-semi-algebraic sets. Furthermore, we define a notion of naive exponential periods and compare it to the existing notions using cohomological methods. This points to a relation between the theory of periods and o-minimal structures.

Theorems & Definitions (186)

• Theorem 1: \ref{['thm:compare_all']}
• Definition 2
• Theorem 3: See Theorem \ref{['thm:naive_is_volume']}
• Theorem 5: Propositions \ref{['prop:coh_implies_naive']} and \ref{['naive_is_effective']}
• Remark 1.1
• Definition 1.2
• Remark 1.3
• Theorem 1.4
• proof
• Theorem 1.5: whitney