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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 sublocal system f R . ... '~" By (4.3.4), I is a constant subfamily 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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 Fall '11
 NormanKatz

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