Dr. Katz DEq Homework Solutions 64

# Dr. Katz DEq Homework Solutions 64 - Then its group of...

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64 N.M. Katz: A polarization of a pure Hodge structure H of weight n is a homo- morphism of Hodge structures (4.2.0.8) (,)H| such that the real bilinear form on H R (4.2.0.9) (2 g i)" (x, C y) is symmetric and positive definite. The positive-definiteness means that (4.2.0.10) if h6H p'q, h+O, then oP-q(2?r i)"(h, h)>0. (4.2.0.11) The category of pure Hodge structures up to isogeny is the category whose objects are pure Hodge structures, but whose morphisms are defined by (4.2.0.12) Hom(up to isogeny)(H, H')=Hom(H, H')| (4.2.0.13) The full subcategory of the category of pure Hodge structures up to isogeny consisting of the polarizable objects (i. e., those which admit at least one polarization) is a semi-simple. It is closed under the formation of internal horn, tensor products, finite direct sums of objects of the same weight, subobjects and quotient objects. Any objects isogenous to a polarizable one is polarizable. (4.2.0.14) Proposition. Let (H, (,)) be a polarized pure Hodge structure.
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Unformatted text preview: Then its group of automorphisms is finite. Proof Any automorphism +p determines an element Up in the group (4.2.0.15) Aut(Hz/torsion)~Aut(Hlt,(2gi)"(x, C y)) which, being the intersection of a discrete and a compact subgroup of Aut (H,), is finite. As ~o-+ Up is clearly a group homomorphism, it remains to prove its kernel is finite. But ~o ~ Up is injective, unless H z = torsion. In the latter case, Aut (Hz) is a finite group. Q.E.D. (4.2.1) Let 5 ~ be a topological space. A family of pure Hodge structures of weight n on 5 e is by defintion a local system H z of Z-modules of finite type, together with a continuously warying filtration F+ i of (Hc)s, the complexification of the stalk of Hz at s, which point by point is a pure Hodge structure of weight n. (4.2.1.1) A polarization of a family of pure Hodge structures of weight n is by definition a morphism of local systems on b ~ (4.2.1.2) (,): Hz|...
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## This note was uploaded on 12/21/2011 for the course MAP 4341 taught by Professor Normankatz during the Fall '11 term at UNF.

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