Jacob Murre

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].

Algebraic Cycles and Algebraic Aspects of Cohomology and K-Theory

These are the notes of the lectures delivered at the C.I.M.E. meeting in Torino, June 93. I have tried to keep the written version as much as possible in the informal style of the lectures. The content of the eight lectures is grouped into seven chapters. In my lectures, with the exception of chapters 4 and 5, the emphasis was on the algebraic methods in studying algebraic cycles; as such the lectures complement those of C. Voisin and M. Green. The chapters are as follows: 1. Algebraic cycles. Basic notions. 2. The Chow ring and the Grothendieck group of coherent sheaves. 3. The Chow ring and higher algebraic K-theory. 4. Introduction to the Deligne-Beilinson cohomology. 5. The Hodge-Conjecture. 6. Applications of the theorem of Merkurjev and Suslin to the theory of algebraic cycles of codimension two. 7. Grothendieck’s theory of motives.

On a Connectedness Theorem for a Birational Transformation at a Simple Point

Introduction. In 1951, Zariski obtained as a special case of his connectedness theorems the fact that the total transform of a normal point by a birational transformation is connected [III]. In a recent paper, [IV], the connectedness of the total transform of a simple point by a birational transformation is proved without using the theory of Fholomorphic functions developed in [III]. In this paper another proof is obtained of this connectedness-theorem for a birational transformation at a simple point. In fact, a somewhat stronger result is proved, namely, the following. Chow has introduced the concept of linear connectedness, a set is said to be linearly connected if every pair of points can be connected by a sequence ofrational curves in that set. In this sense, the total transform of a simple point by a birational transformation is linearly connected.2 Furthermore, some applications are derived for specializations of a complete set of conjugates over a purely transcendental function field. The terminology and notations are from [I]. In conclusion, I want to express my warmest thanks to Professor A. Weil and Professor T. Matsusaka for their advice and interest during the preparation of this paper. I am especially indebted to Professor Weil for suggestions which improve the exposition of the proof of Theorem 1.