The canonical 0-cycle of a K3 surface
November 13, 2013, 3:30 pm
Beauville and I proved that an algebraic K3 surface S has
a 0-cycle which is canonically defined modulo rational equivalence,
and has the property that the intersection of any two divisors
on S is proportional to it. I will review a number of properties
of this cycle, some of which have been discovered by Huybrechts
in his study of spherical objects in the derived category
of S.
On the Chow ring of Calabi-Yau manifolds
November 14, 2013, 3:30pm
I will describe generalizations, some of which are conjectural,
of the canonical ring of a K3 surface to higher dimensional
hyper-Kaehler manifolds or to more general Calabi-Yau manifolds.
For Calabi-Yau hypersurfaces X, for example, I show that
the intersection of any two cycles of complementary nonzero
dimension is proportional to the canonical 0-cycle (the
intersection of a line with X). In the hyper-Kaehler
case, the canonical ring is generated by the divisor classes
and the Chern classes of the tangent bundle and it is conjectured
that the cycle class map is injective on it.
Decomposition of the small diagonal and the topology
of families
November 18, 2013, 3:30pm
The results on the Chow ring of K3 surfaces and of Calabi-Yau
hypersurfaces are obtained by decomposing the
small diagonal in the Chow group of the triple product X
3 . In the case of a K3 surface, this decomposition
has the following consequence on families f : S->B
of projective K3 surfaces parametrized by a quasi-projective
basis B: Up to shrinking B to a dense Zariski open set,
there is a multiplicative decomposition of Rf*Q,
that is a decomposition as the direct sum of its cohomology
sheaves, which is compatible with cup-product on both sides.
This is reminiscent to what happens with families of abelian
varieties, and is very restrictive on the topology of the
family.
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