  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.
