Speaker：Claire Voisin

Time：
October 8th, Tuesday, 9:30-11:30am
October 10th, Thursday, 9:30-11:30am
October 15th, Tuesday, 9:30-11:30am
October 17th, Thursday, 9:30-11:30am

Venue：Quan 9, BICMR

Abstract：
The Chow groups of a smooth projectve variety are very refined invariants whose counterpart in complex geometry is the data of the Hodge structures on cohomology. In fact, there are rather precise conjectures linking both kinds of objects. We will describe these conjectures and explain some technics to attack them in particular cases.

Details:
I. Cycles on algebraic varieties, Chow groups.
I will introduce Chow groups and correspondences between algebraic varieties. I will also discuss cycle classes and cohomological aspects of correspondences (eg the induced morphisms of Hodge structures).

II. Chow groups and coniveau.
I will discuss the relation between 0-cycles and holomorphic forms discovered by Mumford, and explain more generally how assumptions on the size of Chow groups lead to conclusions concerning the size of Hodge structures measured by their coniveau.

III. The Bloch-Beilinson conjectures
This is a fundamental set of conjectures stating conversely that if the cohomology has large coniveau (small niveau) then the Chow groups are small. I will explain many formulations, in particular the nilpotence conjecture, and what can be said on "Bloch-Beilinson's filtrations".

IV. Self-correspondences of surfaces.
I will prove several instances of the Bloch conjecture for self-correspondences acting on 0-cycles of surfaces or threefolds.