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Dr. Katz DEq Homework Solutions 63

# Dr. Katz DEq Homework Solutions 63 - (4.2.0.3...

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Algebraic Solutions of Differential Equations 63 Consequently, the spectral sequence (4.1.2.1) is degenerate at E 2, and the canonical mappings (4.1.2.11) Rqf, (O~/c) ~ E~ q --, E ~ are isomorphisms. Q.E.D. 4.2. Families of Mixed Hodge Structures (4.2.0) We recall that a pure Hodge structure over Z of weight n, H, is by definition, a Z-module of finite type Hz, together with a (Hodge) filtration F of the complex vector space Hcd~ n Hz| which satisfies the condition (4.2.0.0) Hc= ~ FP~ffq(Hc) p+q=n p, qEZ (ffq denotes the complex conjugate of F q. Complex conjugation, and indeed all of Aut(C), acts on H c = H z | through the second factor !). For all pairs (p, q) of integers with p + q = n, we put (4.2.0.1) H p' q = F p n Pq (Hc) , the "subspace of H c of type (p, q)". The bigraduation (4.2.0.2) Hc= 0 HV'q; H q'v=Hv'q p+q=n allows us to define an action s of C* as real group on H c by putting
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Unformatted text preview: (4.2.0.3) s(z)=multiplication by zP~ q on H p'q. Following Well, we put C = s (i). For each z ~C*, the endomorphism s(z) of H c commutes with complex conjugation, and hence comes from an endomorphism still noted s(z) of Hn= Hz| R denoting the field of real numbers. (4.2.0.4) A morphism q9 between pure Hodge structures H and H' is a group homomorphism ~Oz: Hz---~H'z whose C-linear extension <Pc: Hc--* H~: commutes with the action s of C* (4.2.0.3). (Thus between pure Hodge structure of different weights there is only the zero morphism.) (4.2.0.5) The category of pure Hodge structures has an internal Hom and a tensor product | defined in the expected way (cf. [8]). (4.2.0.6) Tate's Hodge structure Z(n) is the rank one Hodge structure of weight - 2 n which is purely of type (- n, - n), and whose integral lattice H z is the subgroup (4.2.0.7) (2n i)" Zc C....
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