Mingmin Shen

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.

On relations among 1-cycles on cubic hypersurfaces

In this paper we give two explicit relations among

Foliations and rational connectedness in positive characteristic

1

THE MOTIVE OF THE HILBERT CUBE

-cycles modulo rational equivalence on a smooth cubic hypersurface

Rationality, universal generation and the integral Hodge conjecture

X

The Motive of the Hilbert Cube X[3]

. Such a relation is given in terms of a (pair of) curve(s) and its secant lines. As the first application, we reprove Paranjapes theorem that

Rational curves on Fano threefolds of Picard number one

\mathrm {CH}_1(X)

Surfaces with involution and Prym constructions

is always generated by lines and that it is isomorphic to

Rational curves on Fermat hypersurfaces

\mathbb{Z}

Prym-Tjurin Constructions on Cubic Hypersurfaces

if the dimension of