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Unformatted text preview: 32 N.M. Katz: is defined (because, in dimension n, all the "lower" ones HW(/), i <a,
h ave source and target the zero module, hence are isomorphisms). In
f act, it is none other than the composite (analogous to (2.3.4.1.3))
1s,na,a
cona.~2 ~.~l~n f [r ~ Jt~, j , t'~X/S) a,na ~ E1 (2.3.6.2)
F ~ ( R "  " f , (f~x/s)) .................................. .+ R""f,(~x/s).
nw(.)
:: (2.3.6.3) Proposition (Igusa, Manin). A ssumptions as in (2.3.6), s uppose S
affine and so small that all the Hodge cohomology sheaves are free (9sm odules. Let co1..... me be a base of R ~ af , (~ax/s) ' a nd denote by (aij) the
m atrix in Me(F(S, (_gs))of H W(a) w ith respect to this base: (2.3.6.4) HW(a) (F~s (coi)) = Z a ji o)j. i (2.3.6.5) C onsider the dual basis o9",...,o9" of R N+anf~(~'~N~sa ). B y
d uality and the definition of the integer a, it is also the least integer with
R N+""f, (12~7s~) nonzero, so that the degeneration of the Hodge =~ De
R ham spectral sequence gives an inclusion
(2.3.6.6) R " +~ "f, ~ (a;,/s). (2.3.6.7) L et ~1 . .... ~ e be Tlinear differential operators on S which are
in the algebra generated by Der(S/T), which operate on the De Rham
c ohomology sheaves Ri f , (Ox/s) via the GaussManin connection 17.Suppose
~
t hat m* .... ,co*, considered as sections of R 2Nn: t ~*x/s1, satisfy the
J,~
d ifferential equation
(2.3.6.8) Z V(~j)(~o*)=0 in Rzu"f,(~2]/s). J T hen each column of the matrix (aij) satisfies the differential equation (2.3.6.9) Z ~j(aji)=0,
J for i=1 ..... (. P roof T he proof is based on the fact that the composite plinear
m apping (2.3.6.9bis) R ""f,(fPx/s) F'~b. , F , s ( R ,  , f , (fPx/s)) n'W(")~ R "~ f , (fPx/s) r
m ay be factored through the subsheaf R , J ,~q2o ~v of h orizontal s ections
x/s)
(el. (2.3.1.3)), as expressed in the commutative diagram ...
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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.
 Fall '11
 NormanKatz

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