Complex Tori

"...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...From what has been said at the beginning of this review, the uniqueness of this first comprehensive 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..." -Zentralblatt Math "Authors have conducted a thorough investigation of these objects. In this volume they present a comprehensive account of their achievements...Containing a great number of new results this monograph will be a standard reference for years to come." --Zentralblatt Math "...general complex tori had a scarce literature so far. This gap is filled by the present book... With its complete up-to-date bibliography the book may serve as both manual and graduate text." --ASM

• 1 Complex Tori.- 1 Homomorphisms of Complex Tori.- 2 Line Bundles.- 3 The Néron-Severi Group.- 4 The Dual Complex Torus.- 5 Extensions of Complex Tori.- 6 Complementary Subtori and Shafarevich Extensions.- 7 Simple and Indecomposable Complex Tori.- 8 The Endomorphism Algebra of a Complex Torus.- 9 The Theorem of Oort and Zarhin.- 10 The Space of all Complex Tori of Dimension g.- 2 Nondegenerate Complex Tori.- 1 Polarizations of Index k.- 2 Moduli Spaces of Nondegenerate Complex Tori.- 3 The Rosati Involution.- 4 The Dual Polarization.- 5 Poincaré’s Reducibility Theorem for Nongenerate Complex Tori.- 6 The Algebraic Dimension.- 7 Picard Number and Algebraic Dimension of Complex Tori of Dimension 2.- 3 Embeddings into Projective Space.- 1 Kähler Theory of Line Bundles on Complex Tori.- 2 Harmonic Forms with Values in a Nondegenerate Line Bundle.- 3 Maps Associated to a Nondegenerate Line Bundle.- 4 The Family of Abelian Varieties Associated to a Nondegenerate Complex Torus.- 5 Some Properties of the Diffeomeorphism ?v_: X ? Xv_.- 6 The Rational C?-map ?L,V_: X ? ?N.- 4 Intermediate Jacobians.- 1 Primitive Cohomology of Kähler Manifolds.- 2 The Griffiths Intermediate Jacobian.- 3 Some Properties of the Griffiths Intermediate Jacobian.- 4 The Weil Intermediate Jacobian.- 5 The Lazzeri Intermediate Jacobian.- 6 The Abelian Variety Associated to the Griffiths Intermediate Jacobian.- 5 Families of Complex Tori.- 1 Complex Tori with Endomorphism Structure.- 2 The Index of a Complex Torus with Endomorphism Structure.- 3 Complex Tori with Endomorphism Structure of Type Ia.- 4 Complex Tori with Endomorphism Structure of Type Ib.- 5 Complex Tori with Endomorphism Structure of Type II.- 6 The Parameter Spaces of Complex Tori with Endomorphism Structure.- 1 The SpaceHg,k.- 2 The Space Km.- 3 The Space Lm,j.- 4 The Space Mm,k.- 5 The Space Nm,k.- 6 The Space Rm,j.- 7 The Space Sn.- 8 The Space Tn.- 9 The Space U(p,q),j.- 10 The Space Vp,q.- 7 Moduli Spaces.- 1 Moduli Problems.- 2 Homogeneous Spaces with Noncompact Stabilizer.- 3 The Moduli Space of all Complex Tori.- 4 Moduli Spaces of Nondegenerate Complex Tori.- 5 Moduli Spaces of Nondegenerate Complex Tori with Endomorphism Structure.- A The Kronecker Product.- B Anti-Involutions on R-Algebras.- C Complex Structures.- References.- Glossary of Notations.