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In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations. Subsequent chapters then develop such topics as Hermitian exterior algebra and the Hodge *-operator, harmonic theory on compact manifolds, differential operators on a Kahler manifold, the Hodge decomposition theorem on compact Kahler manifolds, the Hodge-Riemann bilinear relations on Kahler manifolds, Griffiths's period mapping, quadratic transformations, and Kodaira's vanishing and embedding theorems. The third edition of this standard reference contains a new appendix by Oscar Garcia-Prada which gives an overview of certain developments in the field during the decades since the book first appeared. From reviews of the 2nd Edition: "..the new edition of Professor Wells' book is timely and welcome...an excellent introduction for any mathematician who suspects that complex manifold techniques may be relevant to his work." - Nigel Hitchin, Bulletin of the London Mathematical Society "Its purpose is to present the basics of analysis and geometry on compact complex manifolds, and is already one of the standard sources for this material." - Daniel M. Burns, Jr., Mathematical Reviews.
Complex manifolds. --- Differentiable manifolds. --- Geometry, Algebraic. --- Algebraic geometry --- Geometry --- Differential manifolds --- Manifolds (Mathematics) --- Analytic spaces --- Global analysis (Mathematics). --- Global analysis. --- Analysis. --- Global Analysis and Analysis on Manifolds. --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Mathematical analysis. --- Analysis (Mathematics). --- Manifolds (Mathematics). --- 517.1 Mathematical analysis --- Mathematical analysis --- Geometry, Differential --- Topology
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Moduli theory is the study of how objects, typically in algebraic geometry but sometimes in other areas of mathematics, vary in families and is fundamental to an understanding of the objects themselves. First formalised in the 1960s, it represents a significant topic of modern mathematical research with strong connections to many areas of mathematics (including geometry, topology and number theory) and other disciplines such as theoretical physics. This book, which arose from a programme at the Isaac Newton Institute in Cambridge, is an ideal way for graduate students and more experienced researchers to become acquainted with the wealth of ideas and problems in moduli theory and related areas. The reader will find articles on both fundamental material and cutting-edge research topics, such as: algebraic stacks; BPS states and the P = W conjecture; stability conditions; derived differential geometry; and counting curves in algebraic varieties, all written by leading experts.
Moduli theory. --- Arithmetical algebraic geometry. --- Algebraic geometry, Arithmetical --- Arithmetic algebraic geometry --- Diophantine geometry --- Geometry, Arithmetical algebraic --- Geometry, Diophantine --- Number theory --- Theory of moduli --- Analytic spaces --- Functions of several complex variables --- Geometry, Algebraic --- Geometry, Algebraic. --- Moduli theory --- Algebraic geometry --- Geometry
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