Source: https://language-and-engineering.hatenablog.jp/entry/20140727/HodgeConjecture
Timestamp: 2019-04-25 06:05:40+00:00

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of classes cl(Z) of algebraic cycles.
Two preliminary sections(§1 and 2) introduce the basic ingredients necessary for the statement of the (usual) Hodge conjecture, which we give in §3. In §4 we explain why it must be slightly modified. We treat in §5 the case of codimension 1 cycles (= divisors), due to Lefschetz. In §6 we discuss the few cases where the conjecture is known, and the natural examples where it should be checked. In §7 we explain why the general Hodge conjecture must be modified, as was first noticed by Grothendieck. We examine in §8 some of the consequences of the Hodge conjecture on the structure of the group of algebraic cycles; more generally we say a few words on the relation of Hodge theory with that group.

References: §3
 §4
 §5
 §6
 §7
 §8