Periods of connections and a conjecture of Gross–Deligne

Motivated by a new approach to the Chowla-Selberg formula, Gross and Deligne conjectured at the end of the 70s that periods of geometric Hodge structures with multiplication by an abelian number field are products of special values of the gamma function, with exponents determined by the Hodge decomposition. I will explain a proof of an alternating variant of this conjecture for the cohomology groups of smooth, projective varieties with an automorphism of finite order. The main ingredient is a global-to-local formula for periods of flat vector bundles due to Saito and Terasoma.