How to prove the Hodge conjecture (series unlisted)

Abstract: For a complex manifold or algebraic variety, there are many different invariants one can study. One of the premier options are certain vector spaces called the cohomology of the space. There are many different approaches to cohomology, specialised to different context/purposes: singular, etale, de Rham, crystalline, flat, etc.

There are often ways to take subspaces (called "cycles", in reference to classical homology) and map them into the cohomology groups. In the complex case the ability to integrate over a subspace gives a pairing between the cycles and de Rham cohomology. The Hodge Conjecture states that cohomology classes of a certain kind always arise from cycles. This is closely related to the Tate conjecture, which makes a similar claim for etale cohomology, and more generally to the (conjectural) theory of motives.

In this talk we'll introduce the above ideas in more detail, show how to interpret it in the case of a product of complex elliptic curves, and in fact prove it by explicit computation. This case is actually more generally covered theoretically by the Lefschetz (1,1) theorem, which time permitting we may also discuss.

Mathematics

Audience: general audience

( slides )

BU Community Seminar