This is probably a very stupid question, but could someone explain to me where the weight filtration of mixed Hodge structures come from and why we actually need it? If the Hodgetode Rham spectral sequence degenerates at $E_1$, then why does that not define a pure Hodge structure? I sort of understand that we need this for open varieties, but it seems to me that people talk about MHSs of smooth projective varieties as well. Is there any use of them that does not follow from pure Hodge structures? Thanks!
