Charles Vial

The Fourier transform for certain hyperKähler fourfolds

Using a codimension-1 algebraic cycle obtained from the Poincare line bundle, Beauville defined the Fourier transform on the Chow groups of an abelian variety A and showed that the Fourier transform induces a decomposition of the Chow ring CH∗(A). By using a codimension-2 algebraic cycle representing the Beauville-Bogomolov class, we give evidence for the existence of a similar decomposition for the Chow ring of hyperKahler varieties deformation equivalent to the Hilbert scheme of length-2 subschemes on a K3 surface. We indeed establish the existence of such a decomposition for the Hilbert scheme of length-2 subschemes on a K3 surface and for the variety of lines on a very general cubic fourfold.

Non-commutative resolutions and Grothendieck groups

Let

Exceptional collections, and the Néron–Severi lattice for surfaces

R

Chow–Künneth decomposition for 3- and 4-folds fibred by varieties with trivial Chow group of zero-cycles

be a noetherian normal domain. We investigate when

ON DESCENDING COHOMOLOGY GEOMETRICALLY

R

DISTINGUISHED MODELS OF INTERMEDIATE JACOBIANS

admits a faithful module whose endomorphism ring has finite global dimension. This can be viewed as a non-commutative desingularization of

Parameter Spaces for Algebraic Equivalence

\Spec(R)

Remarks on motives of abelian type

. We show that the existence of such modules forces stringent conditions on the Grothendieck group of finitely generated modules over

PROJECTORS ON THE INTERMEDIATE ALGEBRAIC JACOBIANS

R

Algebraic Cycles and Fibrations

. In some cases those conditions are enough to imply that