On the conjecture of Gross-Deligne
(joint work with V. Maillot)

D. Roessler

Abstract

We prove that the existence of an automorphism of finite order on a $\mathbb{Q}$-variety X implies the existence of non-trivial algebraic linear relations between the logarithms of certain periods of X and the logarithms of special values of the $\Gamma$-function. This implies that a slight variation of results of Andersson, Colmez, Gross and Yosida on the periods of CM abelian varieties are valid for a large class of CM motives, which includes CM abelian varieties. In particular, we prove a weak form of the period conjecture of Gross-Deligne for a certain class of varieties and of Hodge structures cut out in the singular cohomology of the latter.