The decomposition theorem for smooth projective morphisms
says that
decomposes
as .
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
.
We prove however that this is always possible for families of
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 surface
.
We give two proofs of this result, the first one involving
–autocorrespondences
of
surfaces, seen as analogues of isogenies of abelian varieties, the
second one involving a certain decomposition of the small diagonal in
obtained by
Beauville and the author. We also prove an analogue of such a decomposition of the small diagonal
in for Calabi–Yau
hypersurfaces
in ,
which in turn provides strong restrictions on their Chow ring.
Keywords
decomposition theorem, Chow ring, decomposition of the
small diagonal
