Dr. Katz DEq Homework Solutions 71

Dr. Katz DEq Homework Solutions 71 - A lgebraic Solutions...

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Algebraic Solutions of Differential Equations 71 mixed Hodge structures on S an (4.3.5.0) (n n ( O an, Z))san ~ R" 7~, n (Z) has as image a constant sub-family of mixed Hodge structures. Its image is the largest constant sub-local system of R" ft. n (Z). Proof. The first assertion results from the fact that the category of families of mixed Hodge structures is an abelian subcategory of the cate- gory of local systems, and that the formation of images (as well as of kernels and cokernels) commutes with the inclusion of the category of families of mixed Hodge structures into that of local systems. The second assertion will result from the equality a.,~itTO'n-- ~'O,n__ x., 2 in the usual Leray spectral sequence in integral cohomology of rt~": Ua"--~ S an, which itself follows from the following proposition. (4.3.6) Proposition. Hypotheses as in (4.3.5), the Leray spectral sequence Rq an (4.3.6.0) E~'q=HP(S an, ~z, (Z)) ~ HP+q(U an, Z) is degenerate at E 2. It suffices to show that E g'q =0 unless p = 0 or p = 1. This is true, because the R q n."(Z) are local systems on S"", and because S a" is
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