Exponential periods and o-minimality

Let α ∈ ℂ be an exponential period. This paper shows that the real and imaginary parts of α are up to signs volumes of sets definable without parameters in the o-minimal structure generated by 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 parts are up to signs volumes of ℚ -semi-algebraic sets. Furthermore, we define a notion of naive exponential period and compare it to the existing notions using cohomological methods. 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.