Dr. Katz DEq Homework Solutions 69

Dr. Katz DEq Homework Solutions 69 - Algebraic Solutions of...

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Algebraic Solutions of Differential Equations 69 According to the "primitive decomposition" ([43], pp. 77-79), for n < N the mapping (4.3.1.7) t~ Prim.-2if, n(z)(_i) ~L, ,R"f,"(Z) 0 <- i < [n/21 is an isogeny. Thus it remains to polarize Prim"f,"(Z) for n<N. Ac- cording to the Hodge Index Theorem (cf. [43]), the pairing Prim"f," (Z) x Prim"f," (Z) ---* R 2 nf,, (Z) ~ Z (- N) (4.3.1.8) (x,y)--~xwLN-"y, w = cup-product is a polarization. This concludes the proof of (4.3.1.6) and of (4.3.1). (4.3.2) Interpretation. When S=Spec(C), the mixed Hodge structure (4.3.1) on H"(U an, Z) depends only on U, and not on the compactification U~---,X chosen to represent U as the complement in a proper smooth variety X/C of a divisor with normal crossings. This mixed Hodge structure is fimctorial in U, in the sense that if h: U -~ V is a morphism of smooth C-schemes, the induced morphisms h*: H"( V an , Z) -~ H"( U a", Z) are morphisms of mixed Hodge structures. (4.3.2.1) By Hironaka [18], any quasi-projective smooth C-scheme U may be compactified as above, and thus its integral cohomology
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