Dr. Katz DEq Homework Solutions 73

Dr. Katz DEq Homework Solutions 73 - constant. But in that...

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Algebraic Solutions of Differential Equations 73 (4.4.2.3) There exists a finite ~tale covering q~ : S'--~ S such that, for every irreducible Z on the Q-conjugary class A, q~*(P(x) R"f,(O'x/s(log D)), V) is isomorphic to (((gs,) b"(x), d) as a coherent (gs,-module with connection (b,(z) = rank of P(z)(R" f, (Y2~:/s(log D)))). Proof The proof of (4.3.3) applies almost verbatim. The implications (4.4,2.0)~,(4.4.2.1) ~(4.4.2.2) are proved as in (4.3.3), remembering that (P(A) R"~,"(Z), W, F)is polarizable (by (4.4.1) and (4.3.1)). The implica- tion (4.4.2.2)=~(4.4.2.3) is proved as in (4.3.3), remembering that each (P(z) R"f, (Q}/s(log D)), V) has regular singular points, being a sub-object of (R"f, (O}/s (log D)), V). This final implication (4.4.2.3) =~ (4.4.2.1) may be reduced, as in (4.3.3), to the case in which the local system P(A) R" n,"(Z) is
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Unformatted text preview: constant. But in that case, we have (P(A) being idempotent). n an __ (4.4.2.4) P(A) R zc, (Z)- P(A). I where I = the largest constant sub-local system f R . ... '~" By (4.3.4), I is a constant sub-family of mixed Hodge structures, hence (4.4.1) so is P(A)I=P(A)R"~,"(Z), and in particular (4.4.2.1) holds. Q.E.D. 5. Applications to the Question of Grothendieck 5.0. A Global Situation (5.0,0) Let A be a subring of C which is finitely generated over Z, and put T= Spec(A). Let g: S---~ T be a smooth morphism with geometrically connected fibres, and f: X ~ S a projective and smooth morphism. Let D = U Di be a union of divisors in X, each smooth over S, which have normal crossings relative to S. D \ X (5.0.0.0) T (5.0.1) Let q/c S be an affine open neighborhood of the generic point of S over which each of the Hodge cohomology sheaves Re f, (~x/s) is a...
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