Algebraic aspects of the cohomology of complex projective varieties

Poincare duality, the hard Lefschetz theorem, and the Hodge-Riemann relations show that the cohomology rings of smooth complex projective varieties have remarkable properties. Theorems like the decomposition theorem of Beilinson-Bernstein-Deligne-Gabber and results of Fulton-Lazarsfeld and Ross-Toma on positivity properties of Schur classes give further algebraic consequences for cohomology rings of complex projective varieties. I will discuss some purely algebraic analogues of these results. Joint work with Omid Amini, June Huh, and Alan Stapledon.