Periods of Hodge structures and special values of the gamma function

At the end of the 70s, Gross and Deligne conjectured that periods of geometric Hodge structures with multiplication by an abelian number field are products of values of the gamma function at rational arguments, with exponents determined by the Hodge decomposition. We prove an alternating variant of this conjecture for smooth projective varieties acted upon by an automorphism of finite order, thus improving previous results of Maillot and R"ossler. The proof relies on a product formula for periods of regular singular connections due to Saito and Terasoma.