Complex Tori pp 35-62 | Cite as

# Nondegenerate Complex Tori

## 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 Jacobian*J* _{ G } ^{ P } (*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*.

## Keywords

Modulus Space Line Bundle Abelian Variety Chern Class Positive Semidefinite## Preview

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