# Algebraic Geometry and Analysis Geometry by Akira Fujiki, etc., Kazuya Kato, T. Katsura, Y. Kawamata, Y.

By Akira Fujiki, etc., Kazuya Kato, T. Katsura, Y. Kawamata, Y. Miyaoka

This quantity files the court cases of a world convention held in Tokyo, Japan in August 1990 at the topics of algebraic geometry and analytic geometry.

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Therefore the cohomology of with coefficients in an arbitrary L 1 - m o d u l e may be calculated with use of the complex A • 28 A D° A )A where of 4 i ~ (i0) is defined by the matrix which is the transpose of the matrix described above. If A is a thread module then the complex (10) is decomposed into the sum of complexes of the form 0 where •.. generated correspondingly (that is by {£+~[~-Z) ) and D~ IL' is defined by a scalar matrix with the entries (remind that 4 ~ ~'+O)',-l- "=~" ~-" (~+-'q/')--Q-(+-(q/-'~'))) Turn now to the spectral sequence to a filtration in the module tration in the complex 8~A~ from Section 2.

Then Now let 0 aria I" b M ~ I~] ~ ]~-_ ~ X ~- $Z~and I { ~ 5 z~ = b ~S a continuous map of , if I ~ I and ~ I ~ ) = l ~ ~2~has a degree i , if ~ - - I and if ~----I • Then M ~ ( ~ ) = M~(~)~_M~I~)~--N~(%)_~_~z if 4 = - 4 In the next example X = 7 ~and f " T ~ ~< T is a hyperbolic endo- morphism or automorphism. Then M ~ [ { ] : M~({] ~ F({~):N({~)=IL({~)I = =l~6t(E--~)l [12, 18] , where ~ ' ~ ~--~ R ~ is a linear lifting of { Thus M { ( Z ] = ~ { • ( Z ] = < ~ ( Z ) : N{(Z)~--(L~(~. Z ]]tl)~ are the rational functions (here ~----(-I) ~ , where 5~[0{Z(~) such that .

COROLLARY 2. Suppose that ~ is a connected compact aspherical po- 41 H,(X,Z~ is lyhedron and and ~[ X ~ X tOrsion free. Let L(:~{$)::I=O for every ~ 0 is eventually commutative. Then the Nielsen zeta func- tion is a rational one. 3. Polyhedron with the fundamental group Z p THEOREM 2. Let X ( p is prime). be a connected compact polyhedron with ~ ( ~ } ~ Z @ ( p is prime). e. for every cover- H~(~,~--~H~(~, ~and for every ~ 0 . Then the Nielsen zeta function is rational. PROOF. For every ~ 7 0 N ( { ~ ) = ~ C 0 ~ be the generator of Z ~ a n d (~_ {~l~& 0 [2, 12] .