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Deligne's Principle B

Posted by Martin Orr on Thursday, 22 January 2015 at 11:10

As I explained last time, one of the key steps in the the proof of Deligne's theorem on absolute Hodge classes is Principle B. This allows us to take an absolute Hodge classes on one fibre in a family of varieties, and deduce that certain classes are absolute Hodge on other fibres of the same family. In this post I shall explain a proof of Principle B due to Blasius, which I think is simpler than Deligne's original proof.

As I also mentioned last time, one can state Principle B in a number of slightly different forms. I have chosen to use the following version instead of the one I gave last time (where there was a lot hidden in the mention of the Gauss-Manin connection, which I am happy not to have to talk about). This version can be applied to Shimura varieties just as easily, or even more easily, than the previous one.

Theorem. Let \pi \colon \mathcal{A} \to S be a family of abelian varieties over \mathbb{C}, with connected base S. Let v be a global section of R^{2p} \pi_* \mathbb{Q}.

If there is a point s \in S(\mathbb{C}) such that v_{s,dR} is an absolute Hodge class on \mathcal{A}_s, then for every t \in S(\mathbb{C}), v_{t,dR} is an absolute Hodge class on \mathcal{A}_t.

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