Dr. Katz DEq Homework Solutions 67

Dr. Katz DEq Homework Solutions 67 - ( -~...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Algebraic Solutions of Differential Equations 67 ( For each integer n>0, the sheaf R"n,"(Z) on S an is a local system of Z-modules of finite type. As we have seen (4.1.1), the corre- sponding rational local system ( R" n," (Q) = R" n." (Z) | Q is the abutment of the Leray spectral sequence in local systems on S a". ( E~ 'q =RP f, n(Rqj."(Q)) ~ R p+q 7z,n (Q), which is degenerate at E3 (because by (4.1.1) its complexification is). ( As a temporary notational device, let us denote by N i the decreasing filtration of R" rc,"(Q) defined by the spectral sequence ( We define the weight filtration W of the local system R"rc.n(Z) by: ( W/R"rc."(Z)=the inverse image of N2"-iR"r~.n(Q) under the n an canonical mapping R ~ rtan(z)-~ R r~. (Q) (thus W/= 0 if i < n, and W/= all if i > 2 n). The locally free coherent sheaf on S a" g n ~n (C) ~)c (gSan ~--- Rnf: n
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ( -~ R&quot;f, (t2~:/s (log O)) |162 (~s.n is the abutment of the Hodge =:- De Rham spectral sequence E~'q= Rqf,&quot;((t2~/s(log O)) an) ~ RV+qf, n((f2]/s(log O)) a&quot;) 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. ( and [-5]). The corresponding filtration F of R&quot; Tt~&quot; (C) | (gs.~ defines point by point a filtration F~ of the stalk n an ~ n an (R ~, (C))~-R f, ((t~x/s(logD))~&quot;)QCs. .(t~s. ./m~), where m~ denotes the ideal defining the point s6S&quot;&quot;. (4.3.1) Proposition (Deligne-Hodge). The triple (R&quot; rc,&quot;(Z), IV, F) defined above is a polarizable family of mixed Hodge structures on S a&quot;. 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 ( E~'&quot; =RPf,'(R&quot;j, &quot; Z) ~ R p+~ n,&quot; (Z). 5*...
View Full Document

Ask a homework question - tutors are online