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Jan 28, 2017 at 14:33 comment added Donu Arapura To address your question in paragraph 2, if $X$ is any complex variety take the space of Hodge cycles to mean $Hom(\mathbb{Q}(-p), H^{2p}(X,\mathbb{Q})$ in the category of mixed Hodge structures.
Jan 28, 2017 at 14:13 history answered Joe Berner CC BY-SA 3.0