Complex Tori

A lattice in a complex vector space ℂ ^g is by definition a discrete subgroup of maximal rank in ℂ ^g . It is a free abelian group of rank 2g. A complex torus is a quotient X = ℂ ^g / Λ with Λ a lattice in ℂ ^g . A complex torus is a complex manifold of dimension g. It inherits the structure of a complex Lie group from the vector space ℂ ^g . In this chapter we study some properties of complex tori without any additional structure.

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 _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.

A complex torus is an abelian variety if and only if it admits a holomorphic embedding into some projective space. Hence a general complex torus does not admit a projective embedding. We will show in this chapter that if (X, H) is a nondegenerate complex torus of dimension g and index k, then X admits a differentiable embedding into projective space which is holomorphic in g — k variables and antiholomorphic in k variables. For this choose a line bundle L with first Chern class 3H. The vector space H ^k (X,L) is the only nonvanishing cohomology group of L. It may be considered as the vector space of harmonic forms of bidegree (g — k, k) with values in L. Choosing a suitable metric of L, these forms yield the embedding X → ℙ _N . This embedding depends on the choice of a k-dimensional subvector space V of V = T₀X on which the hermitian form H is negative definite. This embedding comes out of the proof of the Riemann-Roch Theorem of [CAV], Chapter 3. It goes back to a trick of Wirtinger [Wi]: A suitable change of the complex structure of X defines in a canonical way a line bundle M which is positive definite and satisfies h ^k (L) = h⁰(M). As we learned from R. R. Simha, this approach appears already in the work of Matsushima (see [Ma]).

The most important examples of nondegenerate complex tori of index k are the intermediate Jacobians of a compact Kähler manifold M. In this chapter we give their definitions, deduce some of their properties and see how they are related. We omit some of their most important aspects, for example the Abel-Jacobi map, which reflects the geometry of the manifold M, since here we are more interested in the complex tori.

The endomorphism algebra End_ℚ(X) of a simple complex torus X is a skew field of finite dimension over ℚ. According to the Theorem of Oort-Zarhin (see Section 1.9) every skew field of finite dimension over ℚ occurs as the endomorphism algebra of a complex torus. For nondegenerate complex tori the situation is completely different: The existence of a polarization H of index k on X gives strong restrictions for End_ℚ(X): The hermitian form H induces an anti-involution ’ on End_ℚ(X). The skew fields F of finite type over ℚ with anti-involution ′ were classified by Albert. In this chapter we work out which of these algebras can be realized as endomorphism algebras of nondegenerate complex tori.

Let P denote the parameter space of one of the families of nondegenerate complex tori with endomorphism structure constructed in Chapter 5. In the special case of abelian varieties, P is a hermitian symmetric space. To be more precise, there are three series of irreducible hermitian symmetric spaces of the noncompact type CI (the Siegel upper half spaces), AIII, and DIII such that any P is a product of members of these (see [Sh] or [CAV], Chapter 9).

In Chapter 5 we constructed families of nondegenated complex tori with endomorphism structure. These families are parametrized by the symetric spaces H_g,_k,K_m, etc. and their products. In Chapter 6 we showed that these parameter spaces are disjoint unions of finitely many flag domains. In particular every such space is of the form H\G with classical group G and H ⊂ G a closed subgroup.