Dr. Katz DEq Homework Solutions 67

# Dr. Katz DEq Homework Solutions 67 - (4.3.0.6-~...

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Algebraic Solutions of Differential Equations 67 (4.3.0.1) 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 (4.3.0.2) R" n," (Q) = R" n." (Z) | Q is the abutment of the Leray spectral sequence in local systems on S a". (4.3.0.3) 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). (4.3.0.4) As a temporary notational device, let us denote by N i the decreasing filtration of R" rc,"(Q) defined by the spectral sequence (4.3.0.3). We define the weight filtration W of the local system R"rc.n(Z) by: (4.3.0.5) 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
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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 (Deligne-Hodge). 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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