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from Part IV - Hodge theoretic invariants of cycles
Published online by Cambridge University Press: 07 May 2010
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 ℙN = {ℂN+1\{0}}/ℂ× be “a” complex projective N-space. A projective algebraic manifold X is a closed embedded submanifold of ℙN. 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 ⊂ ℙN is indeed smooth). The fact that X is projective algebraic implies that X contains ‘plenty’ of subvarieties. Let zk(X) be the free abelian group generated by (irreducible) subvarieties of codimension k in X. If dim X = n, then zk(X) = zn−k (X), the group generated by dimension n − k subvarieties of X.
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