# Deligne's theorem on the Leray spectral sequence and weights

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 ?

• Why do we even need domain specific details? AFAIK the Leray spectral sequence is a special case of the Grothendieck spectral sequence and that one also collapses at $E_2$. – red_trumpet Aug 25 '18 at 16:35
• @red_trumpet : the Grothendieck spectral sequence does not degenerates at $E_2$ in general. You can look at the Hopf fibration for a simple counter-example. This is specific to the Kähler case and is a non-trivial theorem. – student Aug 25 '18 at 17:36
• You do need Hard Lefschetz, but you can find a perhaps somewhat less scary presentation on pp. 467-8 of Griffiths/Harris. (I can't help with your weights question, though.) – Ted Shifrin Aug 25 '18 at 18:39
• @TedShifrin : Thanks for your comment, but I'm only interested by the proof using the weights. I'll edit my question to reflect this. – student Aug 25 '18 at 18:50