A theory of algebraic cocycles (Q1201789)

In the last years some significant progress was made in relating the structure of an algebraic variety \(X\) with its Chow variety Chow\((X)\) [see \textit{H. B. Lawson jun.}, Ann. Math., II. Ser. 129, No. 2, 253-291 (1989; Zbl 0688.14006), \textit{E. Friedlander} Compos. Math. 77, No. 1, 55- 93 (1991; Zbl 0754.14011) and \textit{E. Friedlander} and \textit{B. Mazur}, ``Filtrations on the homology of algebraic varieties'' (Preprint)]. This is the so-called ``bigraded \(L\)-homology theory'' equipped with certain natural operations (which subsumes the standard integral homology groups and which endows them with a natural ``Hodge filtration''). In the paper under review the authors proceed further with this theory by developing a ``cohomology theory'' for an arbitrary complex quasi-projective variety which pairs with the \(L\)-homology theory and carries a ring structure and some fundamental operators. This theory is presented in all details and relies on a number of beautiful and apparently simple ideas. -- In the last part of the paper the relation with the theory of algebraic vector bundles is explained.