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Complex tori. (English) Zbl 0945.14027
Progress in Mathematics (Boston, Mass.). 177. Boston: Birkhäuser. xv, 251 p. (1999).
A complex torus of dimension $$g$$ is a complex manifold $$X$$ which is analytically isomorphic to a quotient manifold $$\mathbb{C}^g/\Lambda$$, where $$\Lambda$$ is a lattice in $$\mathbb{C}^g$$. In other words, the class of complex tori coincides with the class of connected compact complex Lie groups. A complex abelian variety is a complex torus $$X$$ admitting sufficiently many meromorphic functions, which means that $$X$$ is even a smooth projective-algebraic group variety over $$\mathbb{C}$$. Abelian varieties are very special complex tori, since a general complex torus of dimension $$g\geq 2$$ does not admit any meromorphic function different from a constant. Now, whereas the theory of complex abelian varieties has reached a highly advanced stage of its development, after nearly two hundred years of vast and intense research, comparatively few deep results have been known, so far, about the structure and geometry of general complex tori. This is certainly somewhat peculiar in view of the fact that complex tori are amongst the simplest complex manifolds, on the one hand, and of increasing significance (and appearance) in modern algebraic geometry, on the other hand. Even within the theory of abelian varieties, non-algebraic tori inevitably occur as extensions of abelian varieties, as intermediate Jacobians (in the sense of Ph. A. Griffiths), or as Albanese varieties. The book under review is the very first monograph, in the entire mathematical literature, which is exclusively devoted to the systematic and comprehensive study of the structure of arbitrary (i.e., not necessarily algebraic) complex tori. With regard to the afore-mentioned odd discrepancy between the prominent role of complex tori in various branches of mathematics, on the one hand, and the comparatively modest development of an adequate general theory for them in the past, on the other hand, the authors, both well-known as leading experts in the field of complex abelian varieties, have spent many years on the thorough research on this subject. This rewarding undertaking has led to numerous new and deep-going results on the structure of complex tori and, in combination with the recent work of other researchers, to a much better, more systematic understanding of these objects. Most of that material, together with the classical theory, is presented in this comprehensive monograph at issue, and thus available to the mathematical community as a whole. The central topics of the book are those (recently discovered) properties of complex tori, which essentially differ from the corresponding well-known properties of abelian varieties, and this very fact makes the present treatise a unique and particularly valuable piece of the contemporary mathematical literature. – The book consists of seven chapters and three appendices:
Chapter 1: Complex tori; Chapter 2: Nondegenerate complex tori; Chapter 3: Embeddings into projective space; Chapter 4: Intermediate Jacobians; Chapter 5: Families of complex tori; Chapter 6: The parameter spaces of complex tori with endomorphism structure; Chapter 7: Moduli spaces.
Appendix A: The Kronecker product; Appendix B: Anti-involutions on $$R$$-algebras: Appendix C: Complex structures.
The text is enhanced by a list of 32 references, a glossary of notations, and an index of keywords.
Chapter 1 presents the basic notions and classical results on complex tori (e. g., homomorphisms, line bundles, the Néron-Severi group, and dual complex tori), mainly without those proofs that can be found elsewhere, and turns then to the more recent topics such as extensions of complex tori (à la Oort-Zarkhin), simplicity and indecomposability of complex tori, the endomorphism algebra of a complex torus, and analytic families of complex tori.
Chapter 2 deals with complex tori equipped with a nondegenerate Hermitean form of index $$k$$ (i.e., with $$k$$ negative eigenvalues). These objects, which the authors call complex tori with polarization of index $$k$$, are generalizations of polarized abelian varieties, and studied here for the first time. The authors develop a structure theory for these objects, wherever possible in analogy with the corresponding theory of abelian varieties, including a theory of moduli spaces, generalized Rosati involutions, dual polarizations, a modification of Poincarés reducibility theorem, the concept of algebraic dimension for a nondegenerate complex torus, as well as a deeper study of such tori in dimension two.
Chapter 3 is devoted to differentiable embeddings of nondegenerate complex tori of dimension $$g$$ and index $$k$$ into some projective space. This encompasses a suitable Kähler theory for such tori, the study of harmonic forms with values in a nondegenerate line bundle, an important (differentiable) embedding theorem based upon harmonic forms, Wirtinger’s trick and the Riemann-Roch theorem for (associated) abelian varieties, and the study of further differentiably rational maps from nondegenerate polarized complex tori into projective spaces.
Chapter 4 turns to the perhaps most important examples of nondegenerate complex tori of index $$k$$, namely the intermediate Jacobians of compact Kähler manifolds. The material covered here includes the standard theory of the primitive cohomology of a Kähler manifold, the description of Griffiths’s intermediate Jacobian, an outline of A. Weil’s approach to intermediate Jacobians, a comparison to F. Lazzeri’s intermediate Jacobian of a compact oriented Riemann manifold, and a brief discussion of abelian varieties associated with intermediate Jacobians. As the authors were mainly interested in the aspect of complex tori (intermediate Jacobians) occurring here, they have deliberately omitted any digression on the (indeed very important) Abel-Jacobi map.
Chapter 5 introduces the notion of endomorphism structures of a nondegenerate complex torus, and that as a direct generalization of Shimura’s homonymous notion for abelian varieties. Then, in general, the results derived here represent a deep-going extension of Shimura’s theory on families of abelian varieties with endomorphism structure to nondegenerate complex tori with (generalized) endomorphism structure. It turns out that there are essentially three types of endomorphism structures for nondegenerate complex tori of suitable index $$k$$, according to a generalization of A. Albert’s classical structure theorem (1939) for endomorphism algebras of abelian varieties, and the complex tori with those types of endomorphism structure (Ia, Ib, II, respectively) are studied in great detail.
These new and very subtle investigations are continued in the following chapter 6, where the emphasis is put on a thorough description of the parameter spaces for ten different, concrete families of nondegenerate complex tori with endomorphism structure that have been constructed in the foregoing chapter. As these parameter spaces are not necessarily symmetric hermitean spaces any more, their study is based upon a fine analysis of flag domains in certain classical groups, which play an important role in the representation theory of semi-simple Lie groups.
The concluding chapter 7 discusses the further-going question of whether the parameter spaces for the families studied before do actually lead to moduli spaces (in some sense) for the respective moduli problem. It is proved that in two out of three distinguished cases, coarse moduli spaces for certain nondegenerate complex tori with prescribed endomorphism structure do really exist, whereas this is not valid in the remaining third case, where nevertheless the existence of a topological moduli space can be established.
The understanding of the methods and techniques used in the course of chapters 5 and 6 requires a profound knowledge of the Kronecker product of matrices over a ring as well as some familiarity with anti-involutions on real, complex, or quaternionic algebras. The basic facts concerning these prerequisites are compiled in appendix A and appendix B, while the complex structures of some occurring classical Lie groups are explained in appendix C, just for the convenience of the reader.
The entire text is presented in a very systematic, rigorous, clear and complete manner, just as it is typical for these two masterly researchers and authors. As for the prerequisites, the reader is assumed to be familiar with the language of complex algebraic geometry and complex analysis (about à la Griffiths-Harris), and with the classical basic facts on complex tori as presented in the first three chapters of the authors’ previous textbook [H. Lange and C. Birkenhake, “Complex abelian varieties” (1992; Zbl 0779.14012)]. From what has been said at the beginning of this review, the uniqueness of this first comrehensive monograph on general complex tori, together with its excellent arrangement of the presented material, has already made it a standard text on the subject, and therefore it must be seen as both a really needed and a highly welcome complement to the literature on the related areas of mathematics.

##### MSC:
 14K20 Analytic theory of abelian varieties; abelian integrals and differentials 14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry 14K30 Picard schemes, higher Jacobians 32Q99 Complex manifolds