Dr. Katz DEq Homework Solutions 98

Dr. Katz DEq Homework Solutions 98 - y-r.B a , y-iB.dx...

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98 N.M. Katz: The isotypique decomposition of the complex (6.8.1.6) with respect to /~, is given by P()~(-d))(a(n; a, b, c))- d (6.8.1.7) y-e B d Thus , P(g(-d))(A(n; a, b, c)).dx , y- r B. dx =~ y- e- r. . C (2) Ix]- dx. (6.8.1.8) P()~(- f))H~R(X(n; a, b, c)/C(2)) = cokernel of y-e B ~ feB. dx. The gauss-Manin connection V(d/d2) on HIR(X(n;a,b,c)/C(2)) is deduced from the endomorphism of the complex (6.8.1.6) given by Lie(0/O2), where ~?/~2 denotes the unique derivation of A(n;a,b,c) which kills x and extends the derivation d/d2 of C(2). Because 0/~32 commutes with .the action of /1, on A(n; a, b, c), Lie(~?/O2) induces an endomorphism of each of the complexes
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Unformatted text preview: y-r.B a , y-iB.dx which induces the Gauss-Manin connection on the cokernel. (6.8.2) Proposition. Let n, a, b, c be strictly positive integers. Then for every integer {> 1, there is a horizontal morphism of C(2)-modules with connection (cf (6.1.0)) E( (c-'n (a+db+(Cn 1,(a~Ec) C(2) --* P(Z(-f))H~R(X(n; a, b, c)/C (2)) (6.8.2.0) defined by (6.8.2.1) dx eo ~ the class of y--7- el ---, g the class of .-7- Y Proof In concrete terms, the assertion is that the operator (6.8.2.2) (c ( n 1) annihilates the class of dx/y e in H~R(X(n; a, b, c)/C(2)). In fact, direct calculation (cf. [1], p. 60, (11)) shows that in f2~(,:,,b,c)/C(~ ) we have...
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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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