Motivic integration and algebraic cycles

Motivic integration and projective bundle theorem in morphic cohomology

homology and cohomology theory for real projective varieties

Abstract. In this paper we compute Lawson homology groups and semi-topological K-theory for some threefolds and fourfolds. We consider smooth complex projective varieties whose zero cycles are supported on a proper subvariety. Rationally connected varieties are examples of such varieties. The computation makes use of a technique of Bloch and Srinavas, of the Bloch-Kato theorem and of the spectral sequence

homology and cohomology theory for real

Using morphic cohomology, we produce a finite sequence of conjectures, called morphic conjectures, which terminates at the Grothendieck standard conjecture A. We generalize the notion of numerical equivalence and homological equivalence to morphic cohomology and show that these two equivalence relations coincide if and only if the corresponding morphic conjecture is valid. We define morphic signatures and morphic Hodge numbers, and by assuming the corresponding morphic conjecture, we generalize the Hodge index theorem. We prove a conjecture in rational coefficients of Friedlander and Lawson by assuming the Grothendieck standard conjecture B. As a consequence, we prove that the topological filtration from morphic cohomology is equal to the Grothendieck arithmetic filtration for some cases. 1