Masaki Hanamura

Motivic decomposition and intersection Chow groups, I

For a quasiprojective variety S, we define a category CHM(S) of pure Chow motives over S. Assuming conjectures of Grothendieck and Murre, we show that the decomposition theorem holds in CHM(S). As a consequence, the intersection complex of S makes sense as an object IS of CHM(S). In the forthcoming part II we will give an unconditional definition of intersection Chow groups and study some of their properties.

Absolute Chow-Künneth projectors for modular varieties

It is conjectured that Chow-Künneth projectors exist for any X . Note the cohomology classes of Pi are the Künneth components of the diagonal class in H ðX XÞ. So the conjecture is stronger than a conjecture of Grothendieck—Conjecture (C) in his standard conjectures, [Gr], [Kl]—that the Künneth components of the diagonal be classes of algebraic cycles. Further, the projectors Pi should satisfy additional properties, [Mu 2].

Hodge structure on twisted cohomologies and twisted Riemann inequalities. I

We show the twisted cohomology on has a natural polarized Hodge structure and hence derive the analogues of Riemann’s equality and inequailty.