Dr. Katz DEq Homework Solutions 66

Dr. Katz DEq Homework Solutions 66 - strictness, we find...

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66 N.M. Katz: sense that it comes ffom a filtration by sub-local systems of the complexified local system H c. Then there exists a finite Otale covering ~z: Sf'---~ 5 t~ such that the inverse image z~* (Hz, W, F) of (Hz, W, F) on s is a constant family of mixed Hodge structures. Proof Applying (4.2.1.3) to each of the finitely many non-zero gr w, we find a finite &ale covering 7t: 5P'--* 5 ~ on which the gr w become constant families of Hodge structures. So replacing 5 p by 5 '~', it suffices to show that if the gr w are all constant, then H z is constant. The constancy of the gr w signifies that under the action of zq (~ so) on (Hz)~o, for any 7 elq (~ So); (4.2.2.3.0) (1 - ?)(IV, (Hz)so) c ~_ 1 (Hz)so By hypothesis ? preserves the filtrations W and F, hence T and 1-7 are endomorphisms of the mixed Hodge structure ((Hz)~o, W, F). Thus (4.2.2.2), 1-7 is strictly compatible with W. Combining (4.2.2.3.0) and
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Unformatted text preview: strictness, we find (4.2.2.3.1) (1-j(W,)c W,_~ c~image (I-7)=(1-j(W,_0. Since IV, = 0 for n,~ 0, and IV, = all for n >> 0, this implies 1 - ? = 0, and hence Hz is a constant local system. Q.E.D. (4.2.2.4) Remark. Let us agree to call polarizable a family of mixed Hodge structures whose assosciated graded families gr w are all polarizable. Because gr w is exact, finite direct sums, sub-objects and quotient objects of polarizable families of mixed Hodge structures are polarizable. Clearly, any object isogenous to a polarizable one is polarizable. 4.3. Geometric Interpretation (4.3.0) Let T=Spec(C), S a connected smooth C-scheme, f: X~S a projective and smooth S-scheme, and D = U D~ a union of divisors in X which are smooth over S and which cross normally relative to S (4.3.0.0) DC i X~ j ~ U=X-D S 1 T...
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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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