By Christina Birkenhake

ISBN-10: 3662027887

ISBN-13: 9783662027882

ISBN-10: 3662027909

ISBN-13: 9783662027905

This publication explores the speculation of abelian types over the sphere of complicated numbers, explaining either vintage and up to date leads to smooth language. the second one variation provides 5 chapters on fresh effects together with automorphisms and vector bundles on abelian kinds, algebraic cycles and the Hodge conjecture. ". . . way more readable than so much . . . it's also even more complete." Olivier Debarre in Mathematical reports, 1994.

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7 also H vanishes on V2 n iV2 . Since H is nondegenerate, V2 n iV2 = {O} and thus V = V2 + iV2 • This implies the assertion. 0 The hermitian form H is symmetric on V2 , since its imaginary part E vanishes there. 1 the symmetric form B is defined on the whole of V. The following properties of Hand B will be frequently applied. 2) Lemma. a) (H - B)( v, w) = { 2iE(Ov, w) if (v, w) E V (v,w) E V2 X V2 X V . b) If H is positive definite, then Re(H - B) is positive definite on VI' Proof. As for a): H - B = 0 on V x V2 , since H is

3, equa- {J~(v) = e(-nH(v,c) - ~H(c,c)){J~(v+c) for all W E (5) K(L )1. Hence it suffices to prove the Theorem in the case c = o. 56 Chapter 3. ' WN E A(Lh denote a set ofrepresentatives of K(Lh = A(L)J/A I . 3 it suffices to show that the functions e( - ~ B) 'l9fij , v = 1, ... ,N are linearly independent. We will do this by comparing the coefficients of their Fourier series. For all v E V and 1 ~ v ~ N we have according to (2) and (4) e( -~B(v, v))'l9~)v) = aLo (wv' v)-Ie( -~B(v, v) + ~B(v + Wv' v + wJ) .

Hence "~,, is a functor from the category of complex tori into itself. The following proposition says that this functor is exact. 2) Proposition. If 0 --- Xl --- X 2 --- X3 --- 0 is an exact sequence of complex tori, the dual sequence 0 --- X3 --- X2 --- Xl --- 0 is also exact. Proof. Suppose Xv = Vv / Av. Applying the serpent lemma, the induced sequence of lattices 0 --- Al --- A2 --- A3 --- 0 is exact. As a sequence of free abelian groups it splits, so that 0--- Hom(A3' (1\) --- Hom(A 2, (Cl) --- Hom(Au (CI) --- 0 is also exact.

### Complex Abelian Varieties by Christina Birkenhake

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