Nonfiction 4 By David Mumford

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Additional info for Abelian Varieties, Second Edition

Example text

9(z+z,) for some A, a C*. Writing a = z2 - z, this may also be written as 0(z + a) = Ae=(z)B(z) with a fixed A e C*. Making the substitution z r--} z + u (u a U), using the functional equation for B and comparing the multiplicators on both sides, we get that euu) U E U, or irB(a, u) - 1(u) a 2zriZ, u e U. This implies that irH(a, u) - 1(u) = TrH(u, a) -1(u) + vr(H(a, u) H(u, a)) = irH(u, a) -1(u) + 2aiE(a, u) takes only pure imaginary values for all u e V, hence the same holds for 7rH(u, a) - 1(u), and this being complex linear in u, we must have rH(u, a) = l(u) for all u e V.

In fact, the necessity of these conditions is clear. Suppose conversely that they hold. The first implies the existence of a a non-zero homomorphism Og-± M, and the second implies a non-zero homomorphism Ox hence on dualizing, a non-zero T homomorphism M -) Ox. Hence r(a(l)) is a non-zero section of OX, and since % is complete and connected, r(a(1)) is a non-zero scalar. This implies that r o a is an isomorphism, hence a and r are isomorphisms. X x {t}) > 0 and dim H°(X x {t}, L-1 I % x {t}) > 0, and it follows from Corollary 1 that T1 is closed.

Or, in sheaves, ;J = 0 1®02, where D1= Im(#). Moreover, this shows that ¢(M) is A-projective, too, so M splits into the direct sum of Ker(#) and a second submodule isomorphic to ¢(M). Or, in sheaves, . °F2, where (0), 0: -'W" Z-)- k), Now assume (i) holds. Let X be the complex given by the theorem. As in the proof of Corollary 1, dim[Im(dp-1 (& k(y))] and dim[Im(dp (& k(y))] are locally constant. By Lemma 2, applied first to dp: Kr -s Xp+l, and second to dp_1: K"-1 -+ Ker(d,), we get splittings into projective modules: Z,-1®K_1 BP®H,®K, Bp+1®K,+ K, Kp+1 B, is an isomorphism, B, ® H, = Ker(d,), and d,: 1, --).