Motivation : If $f : X \to Y$ is a smooth projective map between algebraic varieties, then there is a theorem by Deligne which says that the Leray spectral sequence degenerates at $E_2$.
The proof I know uses derived categories and Hard Lefschetz on the fibers.
However I understood there was a simpler proof using weights, as suggested in the answer a previous question of mine here.
In the article by Durfee, "A naive guide to mixed Hodge structures", the axioms state that a morphism preserves weights if the morphism is geometric i.e induced by a map of algebraic varieties. However it doesn't seems that the differentials in the Leray spectral sequence are geometric.
Question : Why do the differentials in the Leray spectral sequence preserve the weights ?