By Christina Birkenhake

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

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**Sample text**

2 that this definition coincides with the classical notion of characteristics. Note that a characteristic is only defined for nondegenerate line bundles and determined only up to translations by elements of A(L). Note also that 0 is a characteristic of Lo. 3. As a first application of the notion of characteristics, we extend it to a function V x V ---------+ ~*, also denoted by aL. 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.

J-L E A. Hence the imaginary part 1m g: A --+ IR is a homomorphism extending IR-linearly to a function V --+ IR, denoted by the same symbol. Define a C-linear form l: V --+ C by v f-+ Img(iv) + ilmg(v) . )) 32 Chapter 2. Line Bundles on Complex Tori is a co cycle in Zl(A,Ho(Oy)) equivalent to f. Moreover XL does not depend on v and takes values in

Jk,7r;I) and for all i, j, k E I. 2. ). ). An immediate D computation gives a