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Short version: is the Chow-Künneth motivic decomposition known for $X \hookrightarrow \mathbb{P}^n_k$ a hypersurface over a field $k$?

Long version: let $M(X)$ be the Chow motive of $X$ with rational coefficients. A standard conjecture predicts the existence of projectors $\pi_k$ in $CH^{\dim X}(X \times X)_\mathbb{Q}$, the Chow group with respect to rational equivalence, such that the submotive of $M(X)$ cut off by $\pi_k$ realizes into $H^k(X)$. This is know for very few varieties: essentially projective spaces, curves and surfaces. What about hypersurfaces? I've never seen that written down, but Ayoub uses it in p. 38 of http://user.math.uzh.ch/ayoub/PDF-Files/Leiden.pdf.

I guess the idea is to play with the induced morphism $M(\mathbb{P}^n) \to M(X)$ and the decomposition of $M(\mathbb{P}^n)$. Is there a motivic hyperplane Lefschetz theorem allowing us to say that this is injective?

If anyone has references or knows how to fill in the details I will be very grateful.

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    $\begingroup$ See the proof of lemma 5.1 of arxiv.org/abs/0710.4002 (Iyer-Mueller-Stach, "Chow-Kuenneth decomposition for special varieties"). $\endgroup$ – user31960 Mar 21 '13 at 21:29

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