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We prove that the existence of an automorphism of finite order on a variety X defined over a number field implies the existence of algebraic linear relations between the logarithm of certain periods of X and the logarithm of special values of the Gamma-function. This implies that a slight variation of results by Anderson, Colmez and Gross on the periods of CM abelian varieties is valid for a larger class of CM motives. In particular, we prove a weak form of the period conjecture of Gross-Deligne.
Our proof relies on the arithmetic fixed point formula
(equivariant arithmetic Riemann-Roch theorem)
proved by K. Köhler and the second
author, and the vanishing of the
equivariant analytic torsion for the Dolbeault complex.
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