# Abelian Varieties, Second Edition by David Mumford

By David Mumford

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Assume Y is reduced and connected. (,) HP(X,,, FFy) is a constant function, (i) Rpf. (Jr) is a locally free sheaf & on Y, and for all y e Y, the (ii) natural map - ®m y k(y) > HP(X, ,,) is an isomorphism. If these conditions are fulfilled, we have further that HP-1(Xy, Fr,) RP-' f* (F) ®aYk(y) is an isomorphism for all y e Y. PROOF. Again assume Y affine, S' as in the proposition. (ii) (i) is obvious. To prove (i) LEMMA 1. If Y is reduced and.. (ii), we need two lemmas. a coherent sheaf on Y such that dims[ ° ®0Yk(y)] = r, all y e Y, then JF is locally free of rank ronY.

Then we have: COROLLARY. (a) For each p > 0, the function Y --)- Z defined by ya (b) 9 dimk(v) H1'(Xv, Fv) is upper semicontinuous on Y. The function Y-+ Z defined by (- 1)" dimk(V) H'(X5, Fv) y - X(Fv) = p-0 is locally constant on Y. PRooF. The problem being local on Y, we may assume Y affine. Let K' be a complex as in the proposition; by further localization, we may assume K'to be a free complex. Denote by d": K" K"+l the coboundary of K. We then have p(Kv, ` v) = dimkcv>[ker (d5 ®a k(y))]- dimk(Y)[Im(d"-1 ®A k(y))] = dimk(5)[K"®k(y)] - dimk(v)[Im(dp(& k(y))] -dimk(v)[Im(d"-1® Ic(y))].

Given any complex subtorus Y of X, there is an integer N > 0, and two points x1, x2 a X, xx - x2 e Y such that a section a of a(xi) = 0, a(x2) 0. (1) (2) The hermitian form H is positive definite. (3) The space of holomorphic sections of L®" give an imbedding of X as a closed complex submanifold in a projective space, for each n > 3. PROOF. We have already shown earlier that (1) (2), and (3) u (1) is clear. It remains to assume (2), and deduce (3). We use the expressions "theta-functions for (H, a)" and "section of L(H, a)" interchangeably, making the identifications indicated earlier..