The linear algebra of the decomposition theorem

The decomposition theorem is one of the deepest known facts about the topology of complex projective varieties. Given a map X -> Y of complex projective varieties, with X smooth, it implies strong restrictions on the structure of the cohomology H*(X) as a module over H*(Y). We show that many of these restrictions are linear-algebraic consequences of classically-known properties of H*(X). This enables us to deduce these restrictions in situations where one cannot apply the decomposition theorem, such as in combinatorial Hodge theory and for Chow rings modulo numerical equivalence. Joint work with Omid Amini and June Huh.