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The decomposition theorem for smooth projective morphisms π: X → B
says that Rπ*Q decomposes as ⊕
Riπ*Q[−i]. We describe simple examples
where it is not possible to have such a decomposition compatible with cup
product, even after restriction to Zariski dense open sets of B. We prove
however that this is always possible for families of K3 surfaces (after shrinking
the base), and show how this result relates to a result by Beauville and
the author [J. Algebraic Geom. 13 (2004) 417–426] on the Chow ring of
a K3 surface S. We give two proofs of this result, the first one involving
K–autocorrespondences of K3 surfaces, seen as analogues of isogenies of
abelian varieties, the second one involving a certain decomposition of the
small diagonal in S3 obtained by Beauville and the author. We also prove an
analogue of such a decomposition of the small diagonal in X3 for Calabi–Yau
hypersurfaces X in Pn, which in turn provides strong restrictions on their Chow
ring.
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