  We prove that the existence of an automorphism of finite order on a (defined
over a number field) variety X 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. Koehler and the second author,
and the vanishing of the equivariant analytic torsion for the Dolbeault
complex.
