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Dr. Katz DEq Homework Solutions 15

Dr. Katz DEq Homework Solutions 15 - (1.4.1.7.1 is...

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Algebraic Solutions of Differential Equations 15 As a corollary of (1.3.2) and the definition of the Gauss-Manin con- nection, we have (1.4.1.7) Proposition. Suppose that the Hodge=~De Rham spectral sequence (1.4.1.7.1) EP'~=Rqf,(f2~ls(log D)) =~ RP +q f, (I2~/s(log D)) is degenerate at El, i.e., that gr~ R p + q f, (0~1 s (log D)) = Rq f, (O~/s). Then the associated graded mapping induced by the Gauss-Manin connection is the cup-product with the Kodaira-Spencer mapping (1.3.2.2) p ~ H ~ (S, a~lT | Ri f, (Oer o (X/S))); i.e., the diagram gr p RP+qf, (Q~c/~ (log D)) (1.4.1.7.2) gqf, (f2P/s(log D))- commutes. v > f2~i T | grF p-' RP +q f, (f2~cis (log D)) 7 " > | + % D)) (1.4.1.8) Remark. In case X/S is proper, it follows from Deligne's mixed Hodge theory [8] and a slight modification of his argument ([6], Theo- rem 5.5) that (1.4.1.8.1) If S is any scheme of characteristic zero, the spectral sequence
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Unformatted text preview: (1.4.1.7.1) is degenerate at El, all of its terms E( 'q, E~; ~ are locally free, and its formation commutes with arbitrary change of base S' -~ S. (1.4.1.8.2) If S is any reduced and irreducible scheme whose generic point is of characteristic zero, there exists a non-void Zariski open set q/in S over which the assertions of (1.4.1.8.1) are valid. We conclude this section by stating explicitly a very useful corollary of (1.4.1.7). (1.4.1.9) Corollary. Hypotheses as in (1.4.1.7), fix an integer n>=O, and suppose that M~=R"f,(O~cls(log D)) is an Os-submodule stable under the Gauss-Manin connection, i.e., that (1.4.1.9.1) V(M) c 0~1T | M. (We then say that M is horizontal.) Let us define the induced Hodge filtration of M, Fi(M), by (1.4.1.9.2) F' (M)= M c~ F i R"f, (f2~ls (log D))....
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