Unformatted text preview: (4.3.0.6) ~ R"f, (t2~:/s (log O)) 162 (~s.n is the abutment of the Hodge =: De Rham spectral sequence E~'q= Rqf,"((t2~/s(log O)) an) ~ RV+qf, n((f2]/s(log O)) a") II I Rqf, (f2~/s (log D)) 162 ('0San RP + if* (t2}/s (log D)) 162 (gsa~ which has E1 locally free, and which degenerates at E1 (cf. (1.4.1.8) and [5]). The corresponding filtration F of R" Tt~" (C)  (gs.~ defines point by point a filtration F~ of the stalk n an ~ n an (R ~, (C))~R f, ((t~x/s(logD))~")QCs. .(t~s. ./m~), where m~ denotes the ideal defining the point s6S"". (4.3.1) Proposition (DeligneHodge). The triple (R" rc,"(Z), IV, F) defined above is a polarizable family of mixed Hodge structures on S a". Proof That it is a family of mixed Hodge structures follows from the Deligne's theory, point by point. It remains to see that it is polarizable. Consider the Leray spectral sequence (4.3.1.1) E~'" =RPf,'(R"j, " Z) ~ R p+~ n," (Z). 5*...
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 Fall '11
 NormanKatz
 De Rham cohomology, Sheaf, spectral sequence, Leray spectral sequence, Jean Leray

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