Reference request: Some theorems in an article of Grothendieck.

Question

In "Standard conjectures on algebraic cycles" Grothendieck says:

"The first is an existence assertion for algebraic cycles (considerably weaker than the Tate conjectures), and is inspired by and formally analogous to

Lefschetz's structure theorem on the cohomology of a smooth projective variety over the complex field

The second is a statement of positivity, generalising

Weil's well-known positivity theorem in the theory of abelian varieties. It is formally analogous to the famous Hodge inequalities, and is in fact a consequence of these in characteristic zero."

My question is:

Where can I find (in a book or article) those theorems?

The names(of the theorems) have changed?

Note: My reference request is for books or articles in the current language!

Answer 1

For Lefschetz decomposition: See this article or try Voisin's book. Or try Chap. 7 of Griffth and Harris' "Principles of Algebraic geometry" on Kahler manifolds.

For Weil's positivity theorem: This probably means the theorem that the Rosati involution is positive. This can be found on most books on abelian varieties; choose one depending on whether you need complex varieties, e. g., Lange and Birkenhake, or Mumford, if you need the algebraic proof. Check the contents or index of the books for Rosati involution.