Presents a modern treatment of the theory of theta functions in the context of algebraic geometry.

Consists of three chapters covering the following topics: foundations, bilinear forms and presentations of certain 2-step nilpotent Lie groups, discrete subgroups of the Heisenberg group, the automorphism group of the Heisenberg group, fundamental unitary representations of the Heisenberg group, and the Fourier transform and the Weil-Brezin map.

Abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the one-dimensional ease. The use of theta functions, particularly since Mumford's work, has been an important tool in the study of abelian varieties and invertible sheaves on them. Also, abelian varieties play a significant role in the geometric approach to modern algebraic number theory. In this book, Kempf has focused on the analytic aspects of the geometry of abelian varieties, rather than taking the alternative algebraic or arithmetic points of view. His purpose is to provide an introduction to complex analytic geometry. Thus, he uses Hermitian geometry as much as possible. One distinguishing feature of Kempf's presentation is the systematic use of Mumford's theta group. This allows him to give precise results about the projective ideal of an abelian variety. In its detailed discussion of the cohomology of invertible sheaves, the book incorporates material previously found only in research articles. Also, several examples where abelian varieties arise in various branches of geometry are given as a conclusion of the book.

This book explores the theory of abelian varieties over the field of complex numbers, explaining both classic and recent results in modern language. The second edition adds five chapters on recent results including automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture. ". . . far more readable than most . . . it is also much more complete." Olivier Debarre in Mathematical Reviews, 1994.

This textbook offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry. The emphasis is on geometric constructions over the complex numbers, notably the construction of important classes of abelian varieties and their algebraic cycles. The book begins with complex tori and their line bundles (theta functions), naturally leading to the definition of abelian varieties. After establishing basic properties, the moduli space of abelian varieties is introduced and studied. The next chapters are devoted to the study of the main examples of abelian varieties: Jacobian varieties, abelian surfaces, Albanese and Picard varieties, Prym varieties, and intermediate Jacobians. Subsequently, the Fourier–Mukai transform is introduced and applied to the study of sheaves, and results on Chow groups and the Hodge conjecture are obtained. This book is suitable for use as the main text for a first course on abelian varieties, for instance as a second graduate course in algebraic geometry. The variety of topics and abundant exercises also make it well suited to reading courses. The book provides an accessible reference, not only for students specializing in algebraic geometry but also in related subjects such as number theory, cryptography, mathematical physics, and integrable systems.

The aim of this volume is to bring together research ideas from various fields of mathematics which utilize the heat kernel or heat kernel techniques in their research. The intention of this collection of papers is to broaden productive communication across mathematical sub-disciplines and to provide a vehicle which would allow experts in one field to initiate research with individuals in another field, as well as to give non-experts a resource which can facilitate expanding theirresearch and connecting with others.

This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a course given at the Institut de Mathematiques de Jussieu in 2004 and 2005. Aimed at postgraduate students with a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety. Including notions from other areas, e.g. singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs are given and exercises aid the reader throughout.

This volume contains the proceedings of the conference Recent Advances and New Directions in the Interplay of Noncommutative Algebra and Geometry, held from June 20–24, 2022, at the University of Washington, Seattle, in honor of S. Paul Smith's 65th birthday. The articles reflect the wide interests of Smith and provide researchers and graduate students with an indispensable overview of topics of current interest. Specific fields covered include: noncommutative algebraic geometry, representation theory, Hopf algebras and quantum groups, the elliptic algebras of Feigin and Odesskii, Calabi-Yau algebras, Artin-Schelter regular algebras, deformation theory, and Lie theory. In addition to original research contributions the volume includes an introductory essay reviewing Smith's research contributions in these fields, and several survey articles.

The book contains a complete self-contained introduction to highlights of classical complex analysis. New proofs and some new results are included. All needed notions are developed within the book: with the exception of some basic facts which can be found in the ̄rst volume. There is no comparable treatment in the literature.