By Christina Birkenhake

ISBN-10: 3662027887

ISBN-13: 9783662027882

ISBN-10: 3662027909

ISBN-13: 9783662027905

This ebook explores the speculation of abelian kinds over the sphere of advanced numbers, explaining either vintage and up to date leads to smooth language. the second one version 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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**Extra info for Complex Abelian Varieties**

**Example text**

This will turn out to be very useful for later computations in particular for the theory of Chapter 6. Suppose L = L(H, X) is a nondegenerate line bundle on X and c is a characteristic for L with respect to the decomposition V = VI EB V2. Define aL: V x V ---------+ (C* by aL(u,v) = Xo(u)e(21fiE(c,u) +1fH(v,u) + ~H(u,u)) . 3. So aL is in fact an extension to V x V of the canonical factor of L. Certainly aL: V x V ---------+ ~* is not a factor of automorphy. The following technical lemma gives some properties of a L' which turn out to be very useful subsequently.

O This implies the assertion. The bilinear form B enables us to introduce the classical factor of automorphy for L in a coordinate free way. Define e L: A x V ---+

The theorem was proven for dimension 2 by Humbert [1] applying a result of Appell [1] and by Lefschetz [1] in general. The present formulation appears in Weil [3] and Mumford [2]. Holomorphic line bundles on a complex torus can be described in terms of factors of automorphy. This construction is basic for this chapter. We recall it in Appendix B for the convenience of the reader. An alternative statement of the Appell-Humbert Theorem is as follows: there is a canonical way to associate to any line bundle L on X a factor of automorphy.

### Complex Abelian Varieties by Christina Birkenhake

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