Math Seminars.
Vangelis Mouroukos (U of M),
Arithmetic Hodge Structures and Higher Abel-Jacobi Maps
(after Green, Asakura and Saito)
One of the traditional tools for the Hodge-theoretic study of
algebraic cycles has been the Abel-Jacobi map, classically for
divisors on Riemann surfaces and generalized by Weil and
Griffiths to higher codimension cycles on varieties of
arbitrary dimension. However, both the image and the kernel of
the Abel-Jacobi map remain, in general, rather mysterious.
Recent work by various authors associates to any complex
variety a new Hodge-theoretic structure, called an arithmetic
Hodge structure. Using these, one can define higher Abel-Jacobi
maps from (higher) Chow groups to certain extension groups of
arithmetic Hodge structures. These new invariants can be used
to study the kernel and image of the Abel-Jacobi map in several
interesting situations (for example, in the cases of zero-cycles
on a surface and cycles on general hypersurfaces). We shall
give an exposition of the above constructions and examples to
illustrate that the new invariants can detect non-trivial
algebraic cycles quite efficiently.