6 - Three lectures on the Hodge conjecture

Summary

Abstract

The statement of the Hodge conjecture for projective algebraic manifolds is presented in its classical form, as well as the general (Grothendieck amended) version. The intent of these lectures is to focus on some specific examples, rather than present a general survey overview, as can be found in [Lew2] and [Shi]. A number of exercises for the reader are sprinkled throughout the lectures. For background material, the reader is assumed to have some familiarity with the geometry of complex manifolds, such as can be found in chapter 0 of [G-H1].

Keywords: Hodge conjecture, normal function, Abel–Jacobi map, algebraic cycle. 1991 Mathematics subject classification: 14C30, 14C25

Lecture 1: the statement and some standard examples

Some preliminary material

Let ℙ^N = {ℂ^N+1\{0}}/ℂ^× be “a” complex projective N-space. A projective algebraic manifold X is a closed embedded submanifold of ℙ^N. By a theorem of Chow, X is cut out by the zeros of a finite number of homogeneous polynomials, satisfying a certain jacobian criterion (so that X ⊂ ℙ^N is indeed smooth). The fact that X is projective algebraic implies that X contains ‘plenty’ of subvarieties. Let z^k(X) be the free abelian group generated by (irreducible) subvarieties of codimension k in X. If dim X = n, then z^k(X) = z_n−k (X), the group generated by dimension n − k subvarieties of X.