Arithmetic Study Group: Higher Abel-Jacobi maps
9 August 2005 00:00 in CM219
"This talk will be concerned with detecting cycles modulo rational equivalence on a (complex) projective algebraic variety -- that is, elements in the Chow group. We will stick to zero-cycles (Q-linear combinations of points) for simplicity. The work described builds on that of Green, Griffiths, Lewis, Voisin and others.
The problem is completely solved by Abel's theorem when the variety is a curve. But already for a surface of positive geometric genus, Mumford's theorem says that the kernel of the Albanese map (i.e. the Abel-Jacobi map on 0-cycles) is "huge". The problem of detecting this kernel leads us to consider "spreads" of cycles which take into account their field of definition. The present talk will be devoted to explaining how a filtration on the Chow group and "higher" Abel-Jacobi maps emerge from this construction. (Our approach is to avoid actually "using" the arithmetic Bloch-Beilinson conjecture or the Hodge conjecture.)
We will conclude by describing some applications to 0-cycles on products of curves, where there are links to regulators on algebraic K-theory, iterated integrals, and transcendental number theory (and of course some beautiful applications of Hodge theory). "
Note the unusual room: this talk will take place in CM221.