Abstract
To any smooth projective curve C one can associate an abelian variety, the Jacobian variety J(C). In [W2] Weil showed that, more generally, to any smooth projective variety M of dimension n and any p ≤ n, one can associate an abelian variety, the p-th intermediate Jacobian of M. It has, however, the disadvantage that it does not depend holomorphically on M in general. It was Griffiths’ idea to modify the definition in such a way that the new intermediate JacobianJ P G (M) varies holomorphically on with M. It is a complex torus, but in general not an abelian variety. It admits, however, a class of line bundles whose first Chern class is a nondegenerate hermitian form. This is a special case of the following situation: Let X be a complex torus of dimension g and H ∈ NS(X) a nondegenerate hermitian form. Suppose k denotes the index of H, that is, the number of negative eigenvalues of H. We call such a hermitian form a polarization of index k. (Note that in [G] H is called a k-convex polarization). If H is a polarization of index k on a complex torus X, we call the pair (X, H) a nondegenerate complex torus of index k. In view of the definition of a pseudo-Riemannian manifold [He] one might be tempted to call (X, H) a pseudo-abelian or semi-abelian variety, but these notions have already a different meaning. Note that a nondegenerate complex torus of index 0 is a polarized abelian variety. It is the aim of this chapter to derive the main properties of nondegenerate complex tori of index k.
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© 1999 Springer Science+Business Media New York
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Birkenhake, C., Lange, H. (1999). Nondegenerate Complex Tori. In: Complex Tori. Progress in Mathematics, vol 177. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1566-0_2
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DOI: https://doi.org/10.1007/978-1-4612-1566-0_2
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7195-6
Online ISBN: 978-1-4612-1566-0
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