Dr. Katz DEq Homework Solutions 30

Dr. Katz DEq Homework Solutions 30 - 30 N.M. Katz: is a s...

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Unformatted text preview: 30 N.M. Katz: is a s ection o f the inclusion F~= i r ~-1, whose kernel is none other t hen F i+lnF~o~ ~-1. T his shows that the canonical mapping F~+lc~ F~o~i - 1 ~ F~o~i-a/Ffo~iis a n isomorphism, which proves (2.3.4.4.2). (2.3.5) In this section we wish to explain the differential equations satisfied by certain of the higher Hasse-Witt matrices introduced in (2.3.4). These differential equations were first noticed by Igusa [22] in t he case of elliptic curves, then later explained quite generally by Manin [28]. (2.3.5.1) We place ourselves under the hypotheses of (2.3.2), and assume f urther that the divisor D is void, and that the geometric fibres of X /S a re c onnected, of dimension N. Under these hypotheses, we have (2.3.5.1.1) R2Nf,(~Q~c/s)--%RNf,(O~/s) is locally free of rank 1 and even c anonically isomorphic to the structural sheaf Cs, via the trace morphism (cf. [15] and [42]) tr: RaNf , (Y2]/S)~ d)s (2.3.5.1.2) w hich carries the Gauss-Manin connection on R ZNf,(f2]/s) i nto the s tandard connection on d)s (the one given by exterior differentiation d : (9s'-~ 12~/r). T he cup-product pairings (2.3.5.1.3) Rn f , ([2*x/s)@ R 2N-'f, (fd~c/s)~ tr R2N f (Q~c/s) " ' (gs a re perfect dualities of coherent locally free S modules, for which the f iltrations F and F~o~are both self-dual in the sense that (2.3.5.1.4) ( F' R n f , (O~/s)) • = F N +1 -n R a N - nf , (f2~/s) (2.3.5.1.5) (F~'onR ' f , (O~c/s))• = F~o, ~-n RaN-n f , (12~;/s). + T he associated graded pairings N--i 2N-n r iR n 12* f2* g F( f*(X/S)) | gF (R f,(x/s) n-i i N+i-n ~ N 2N 9 r gF( R f,( [2 s))) N-i a nd , grfoo. (R2Sf, (f2~:/s)) (2.3.5.1.7) F ~* i ,bs(Rn - - i f ,(~x/s))| /l /l * N+i-n N-f,(f2x/si )) ~ * N . F.bsR N f,(f2x/s) "" __~(9 s ...
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