A canonical arithmetic quotient for actions of lattices in by David Fisher

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[~i n+l , I] '''''Zn+l [~i n+l' - 49 - The orientation of the local coordinating agrees with the natural orientation of CP(n + i) if and only if n + 1 is even, hence the sign (-i) n+l. On the other hand the normal bundle to CP(n) is ~ > CP(n) and in each fibre the repret sentation of Z is multiplication by 2 . pS 9 Now surely Tr([CP(n + l)~Z ]) = ind [CP(n + i)] = 0 or 1 pS according to whether n is even or odd. Since Tr[CP(n + I),Z pS ] = Lc(X(j, n + i)) + (-l)n+l(Lc(X(j, i))) n+l the lemma follows.

51 - This is split into a direct sum hl,O ~ h O' i of complex subspaces where h I'O is the space of holomorphic 1-forms, and hO~I the space of anti-holomorphic 1-forms. An isomorphism of (HI(M2;R),J) with h0'I is defined as follows. o<. o<). If (T~M 2) is a holomorphic map of finite period we shall focus our attention on the trace of (T*,h0'I) because this is conjugate 'to (T*,hl'O)and Tr[T*,M 2] = tr(T*,hO, I) - tr(T*,hl,O). We fix an odd integer n > 1 and set ~ = exp (2~-i/n). a In CP(2) we introduce the non-singular algebraic curve S = Since , z2, + z2 + z3 = O .

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