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Hello to all of you : I would like to know if it is true that the Grothendieck period conjecture implies the Hodge conjecture in the case of non-singular complex algebraic varieties ? Can you explain to me why and how ? Thanks a lot.

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What makes you think the period conjecture implies Hodge? The two conjectures can be phrased by saying that the realization functor from Chow motives to a) Hodge structures and b) triples $(V_1, V_2, \omega)$ with $V_1$, $V_2$ $\mathbf Q$-vsp. and an iso $\omega: V_1 \otimes \mathbf C \r V_2 \otimes \mathbf C$ are full (i.e. surjection on morphisms). One of the two conj. would imply the other e.g. if there was a functor between these categories, but I don't see that functor. However, both the period and Hodge conj. imply the standard conj. All of this is in Andre's book on motives, ch. 7. –  Jakob May 25 '13 at 3:20
I meant to say: one would imply the other if there was a full functor (but I don't even see the existence of the functor in either direction). –  Jakob May 25 '13 at 3:22

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