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Skoda–El Mir Theorem
The Skoda–El Mir theorem is a theorem of complex geometry, stated as follows: Theorem ( Skoda, El Mir, SibonyN. Sibony, ''Quelques problemes de prolongement de courants en analyse complexe,'' Duke Math. J., 52 (1985), 157–197). Let ''X'' be a complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such th ..., and ''E'' a closed complete pluripolar set in ''X''. Consider a closed positive current \Theta on X \backslash E which is locally integrable around ''E''. Then the trivial extension of \Theta to ''X'' is closed on ''X''. Notes References * J.-P. Demailly,'L² vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)' Complex manifolds Several c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Geometry
In mathematics, complex geometry is the study of geometry, geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of space (mathematics), spaces such as complex manifolds and Complex algebraic variety, complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and coherent sheaf, coherent sheaves. Application of transcendental methods to algebraic geometry falls in this category, together with more geometric aspects of complex analysis. Complex geometry sits at the intersection of algebraic geometry, differential geometry, and complex analysis, and uses tools from all three areas. Because of the blend of techniques and ideas from various areas, problems in complex geometry are often more tractable or concrete than in general. For example, the classification of complex manifolds and complex algebraic varieties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henri Skoda
Henri Skoda (born 1945) is a French mathematician, specializing in the analysis of several complex variables. Skoda studied from 1964 at l'École normale supérieure and received there in 1967 his agrégation in mathematics. He received in 1972 his Ph.D. from the University of Nice Sophia Antipolis under André Martineau (primary advisor) and Pierre Lelong (secondary advisor) with thesis ''Étude quantitative des sous-ensembles analytiques de Cn et des idéaux de functions holomorphes''. Skoda became a professor in Toulon and since 1976 has been a professor at the University of Paris VI. For many years he ran an analysis seminar with Pierre Lelong and Pierre Dolbeault. In 1978 Skoda received the Poncelet Prize and as an invited speaker at the International Congress of Mathematicians in Helsinki Helsinki () is the Capital city, capital and most populous List of cities and towns in Finland, city in Finland. It is on the shore of the Gulf of Finland and is the seat of southern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nessim Sibony
Nessim Sibony (20 October 1947 – 30 October 2021) was a French mathematician, specializing in the theory of several complex variables and complex dynamics in higher dimension. Since 1981, he was professor at the University of Paris-Sud in Orsay. Biography Sibony received in 1974 his PhD from the University of Paris-Sud with thesis ''Problèmes de prolongement analytique et d'approximation polynômiale pondérée''. His research deals with complex analysis and complex dynamics in several variables, including collaboration with John Erik Fornæss and Dinh Tien-Cuong on Fatou-Julia theory in several complex variables and on singular foliations by Riemann surfaces. Independently of Adrien Douady and John H. Hubbard, Sibony proved in the 1980s that the Mandelbrot set is connected. In 1985 he received the Vaillant Prize and in 2009 the Sophie Germain Prize from the French Academy of Sciences. For 2017 he received the Stefan Bergman Prize. In 1990 he was an Invited Speaker with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifold
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas (topology), atlas of chart (topology), charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are Holomorphic function, holomorphic. The term "complex manifold" is variously used to mean a complex manifold in the sense above (which can be specified as an ''integrable'' complex manifold) or an almost complex manifold, ''almost'' complex manifold. Implications of complex structure Since holomorphic functions are much more rigid than smooth functions, the theories of smooth manifold, smooth and complex manifolds have very different flavors: compact space, compact complex manifolds are much closer to algebraic variety, algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth ''n''-dimensional manifold can be Embedding, embedded as a smooth subma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pluripolar Set
In mathematics, in the area of potential theory, a pluripolar set is the analog of a polar set for plurisubharmonic functions. Definition Let G \subset ^n and let f \colon G \to \cup \ be a plurisubharmonic function which is not identically -\infty. The set : := \ is called a ''complete pluripolar set''. A ''pluripolar set'' is any subset of a complete pluripolar set. Pluripolar sets are of Hausdorff dimension at most 2n-2 and have zero Lebesgue measure. If f is a holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ... then \log , f , is a plurisubharmonic function. The zero set of f is then a pluripolar set if f is not the zero function. See also * Skoda-El Mir theorem References *Steven G. Krantz. ''Function Theory of Several Complex Variables'', AMS Ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Current
In mathematics, more particularly in complex geometry, algebraic geometry and complex analysis, a positive current is a positive (''n-p'',''n-p'')-form over an ''n''-dimensional complex manifold, taking values in distributions. For a formal definition, consider a manifold ''M''. Currents on ''M'' are (by definition) differential forms with coefficients in distributions; integrating over ''M'', we may consider currents as "currents of integration", that is, functionals :\eta \mapsto \int_M \eta\wedge \rho on smooth forms with compact support. This way, currents are considered as elements in the dual space to the space \Lambda_c^*(M) of forms with compact support. Now, let ''M'' be a complex manifold. The Hodge decomposition \Lambda^i(M)=\bigoplus_\Lambda^(M) is defined on currents, in a natural way, the ''(p,q)''-currents being functionals on \Lambda_c^(M). A positive current is defined as a real current of Hodge type ''(p,p)'', taking non-negative values on all positive ''(p,p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Demailly
Jean-Pierre Demailly (25 September 1957 – 17 March 2022) was a French mathematician who worked in complex geometry. He was a professor at Université Grenoble Alpes and a permanent member of the French Academy of Sciences. Early life and education Demailly was born on 25 September 1957 in Péronne, France. He attended the Lycée de Péronne from 1966 to 1973 and the Lycée Faidherbe from 1973 to 1975. He entered the École Normale Supérieure in 1975, where he received his agrégation in 1977 and graduated in 1979. During this time, he received an undergraduate ''licence'' degree from Paris Diderot University in 1976 and a ''diplôme d'études approfondies'' under Henri Skoda at the Pierre and Marie Curie University in 1979. He received his ''Doctorat d'État'' in 1982 under the direction of Skoda at the Pierre and Marie Curie University, with thesis "Sur différents aspects de la positivité en analyse complexe". Career Demailly became a professor at Université Grenoble Alp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifolds
In differential geometry and complex geometry, a complex manifold is a manifold with a ''complex structure'', that is an atlas of charts to the open unit disc in the complex coordinate space \mathbb^n, such that the transition maps are holomorphic. The term "complex manifold" is variously used to mean a complex manifold in the sense above (which can be specified as an ''integrable'' complex manifold) or an ''almost'' complex manifold. Implications of complex structure Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth ''n''-dimensional manifold can be embedded as a smooth submanifold of R2''n'', whereas it is "rare" for a complex manifold to have a holomorphic embedding into C''n''. Consider for example any com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Several Complex Variables
The theory of functions of several complex variables is the branch of mathematics dealing with functions defined on the complex coordinate space \mathbb C^n, that is, -tuples of complex numbers. The name of the field dealing with the properties of these functions is called several complex variables (and analytic space), which the Mathematics Subject Classification has as a top-level heading. As in complex analysis of functions of one variable, which is the case , the functions studied are '' holomorphic'' or ''complex analytic'' so that, locally, they are power series in the variables . Equivalently, they are locally uniform limits of polynomials; or locally square-integrable solutions to the -dimensional Cauchy–Riemann equations. For one complex variable, every domainThat is an open connected subset. (D \subset \mathbb C), is the domain of holomorphy of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
