Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon-Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry.This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the one-dimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.
Bibliographic references
Includes bibliographical references and index.
Source of description
Description based on print version record.
Contents
Cover
Title page
Contents
Preface
Chapter 1. Residue calculus in one variable
1.1. An analytic approach
1.1.1. Laurent expansions and meromorphic functions
1.1.2. Residue of a (1,0) meromorphic form
1.1.3. Jordan's lemma and Mellin transform
1.2. The geometric point of a view
1.2.1. Factorization of integration currents through residue currents
1.2.2. Residue currents and Lagrange's interpolation formula
1.2.3. Residue currents and Bergman-Weil developments
1.2.4. Residue currents and multiplicative calculus
1.2.5. Residues of sections of hermitian bundles
1.3. The algebraic point of view
1.3.1. Cauchy's formula in ¹_{\C}=\C or \P¹_{\C} and division in \K[ ]
1.3.2. Currential Cauchy's formula on compact Riemann surfaces
1.3.3. Green currents and heights of arithmetic cycles
1.3.4. Residues, residue currents, and Gorenstein \K-algebras
Chapter 2. Residue currents: a multiplicative approach
B.1.1. Cocycles on a complex analytic space, examples
B.1.2. Linear algebra and construction of cocycles
B.1.3. Holomorphic bundles over a complex analytic space
B.1.4. Isomorphism classes of holomorphic bundles
B.1.5. Local holomorphic frames and sheaves of holomorphic sections
B.1.6. The concepts of local versus global complete intersection
B.1.7. The bundles ^{*( , )}_{\X} over a complex manifold \X
B.2. Sheaves of bundle-valued differential forms or currents
B.2.1. Differential forms and currents on a complex manifold.
B.2.2. Differential forms and currents on a complex analytic space.
ISBN
1-4704-7489-1
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