By Christina Birkenhake
This publication explores the idea of abelian forms over the sphere of complicated numbers, explaining either vintage and up to date ends up in sleek language. the second one version provides 5 chapters on fresh effects together with automorphisms and vector bundles on abelian forms, algebraic cycles and the Hodge conjecture. ". . . way more readable than such a lot . . . it's also even more complete." Olivier Debarre in Mathematical experiences, 1994.
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Extra resources for Complex Abelian Varieties
This proves the assertion. 2 we associated to every holomorphic line bundle on X an alternating Z-valued form on A and thus via JR-linear extension an alternating form V x V --t JR. Conversely, we will determine, which alternating forms come from line bundles in this way. 6) Proposition. For an alternating form E: V x V --t JR the following conditions are equivalent: i) There is a holomorphic line bundle L on X such that E represents the first Chern class c1 (L ) . ii) E(A, A) ~ Z and E(iv,iw) = E(v,w) for all v,w E V.
For all u = u l + u2,v = VI + v2, W E V = VI EB V2 a) adu,v + w) = aL(u,v)e(1fH(w,u)), b) aL(u + v, w) = aL(u, v + w)aL(v, w)e(21fiE(u l , V2))' c) adu,v)-I = aL(-u,v)Xo(u)-2 e(-1fH(u,u)), d) aL'(u,v) = aL(u,v)e(21fiE(w,u)) with L' = t':nL. Proof. 2 respectively. 1: aL(u+v,W) = = Xo(u + v)e(21fiE(c, u + v) = + 1fH(w, u + v) + ~H(u + v, u + v)) Xo(u)e(21fiE(c,u) + 1fH(v + w,u) + ~H(u,u)) . Xo(v)e(21fiE(c, v) + 1fH(w, v) + ~H(v, v)) . e(1fiE(u,v) - 21fiE(U2' VI) = + ~H(u,v) - aL(u,v+w)adv,w)e(21fiE(UI,V2)) .
Then c1(/* M) = H, and thus L ® /* M- 1 E Pic°(X1). 3 the homomorphism /*: Pic°(X2) ---- Pic°(X1) is surjective, so there is an N E Pic°(X2) with /* N = L ® /* M- 1. Now M = M ® N satisfies i). 4. D Let L be a line bundle on X. For any point x E X the line bundle t;L®L- 1 has first Chern class zero. 3 is a homomorphism. In order to compute its analytic representation, suppose L = L(H, X) and X = VIA. 5) Lemma. The map
Complex Abelian Varieties by Christina Birkenhake