Now I want to study some fundamental theorems of cohomology of (complex) manifolds. e.g., Poincare duality, Kunneth formula, weak and hard Lefschetz, cup product, cycle maps, "de Rham cohomology = singular cohomology = derived sheaf cohomology = Cech cohomology", and so on. I'm reading Bott-Tu's "differential forms in algebraic topology", so I know some of these for De Rham cohomology. I want to know these theorems for integers coefficient cohomology.
Glancing through the table of contents, it seems that the chapter 0 of Griffiths-Harris contains all of these (for complex manifolds). But many people say that it is too brief for the people who study it first time. (I've read Hartshorne, so I think it may be readable for me...)
So please suggest to me some references.
Thank you very much.