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From SFB1085 - Higher Invariants
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Let α ∈ ℂ be an exponential period. In this talk I will present on joint work with Philipp Habegger and Annette Huber. 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 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 ℚ-semialgebraic sets; and it points to a relation between the theory of periods and o-minimal structures. No prior knowledge of o-minimality is assumed.

Furthermore, we compare the definition of naive exponential periods to the existing definitions of cohomological exponential periods and periods of Nori motives and show that they all lead to the same notion. 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.

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