June 26, 2022

Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Coxeter Lecture Series
November 13,14,18, 2013 at 3:30 p.m.
Fields Institute, Room 230

Claire Voisin
Institute de M
athématiques de Jussieu

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.

Speakers in the Distinguished and Coxeter Lecture Series have made outstanding contributions to their field of mathematics. The Lecture Sereies consists of three one-hour lectures.

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