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Positive Form
In complex geometry, the term ''positive form'' refers to several classes of real differential forms of Hodge type ''(p, p)''. (1,1)-forms Real (''p'',''p'')-forms on a complex manifold ''M'' are forms which are of type (''p'',''p'') and real, that is, lie in the intersection \Lambda^(M)\cap \Lambda^(M,). A real (1,1)-form \omega is called semi-positive (sometimes just ''positive''), respectively, positive (or ''positive definite'') if any of the following equivalent conditions holds: #-\omega is the imaginary part of a positive semidefinite (respectively, positive definite) Hermitian form. #For some basis dz_1, ... dz_n in the space \Lambda^M of (1,0)-forms, \sqrt\omega can be written diagonally, as \sqrt\omega = \sum_i \alpha_i dz_i\wedge d\bar z_i, with \alpha_i real and non-negative (respectively, positive). #For any (1,0)-tangent vector v\in T^M, -\sqrt\omega(v, \bar v) \geq 0 (respectively, >0). #For any real tangent vector v\in TM, \omega(v, I(v)) \geq 0 (respectively, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Geometry
In mathematics, complex geometry is the study of geometric structures and constructions arising out of, or described by, the complex numbers. In particular, complex geometry is concerned with the study of spaces such as complex manifolds and complex algebraic varieties, functions of several complex variables, and holomorphic constructions such as holomorphic vector bundles and 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 through the minimal model program and the construction of moduli spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Surface
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * Complex (album), ''Complex'' (album), by Montaigne, 2019, and its title track * Complex (EP), ''Complex'' (EP), by Rifle Sport, 1985 * Complex (song), "Complex" (song), by Gary Numan, 1979 * Complex Networks, publisher of magazine ''Complex'', now online Biology * Protein–ligand complex, a complex of a protein bound with a ligand * Exosome complex, a multi-protein intracellular complex * Protein complex, a group of two or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifolds
In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \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), and 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 compact connected complex manifold ''M'': any holomorphic function on it is cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...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 Grenobl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joe Harris (mathematician)
Joseph Daniel Harris (born August 17, 1951) is a mathematician at Harvard University working in the field of algebraic geometry. After earning an AB from Harvard College, he continued at Harvard to study for a PhD under Phillip Griffiths. Work During the 1980s, he was on the faculty of Brown University, moving to Harvard around 1988. He served as chair of the department at Harvard from 2002 to 2005. His work is characterized by its classical geometric flavor: he has claimed that nothing he thinks about could not have been imagined by the Italian geometers of the late 19th and early 20th centuries, and that if he has had greater success than them, it is because he has access to better tools. Harris is well known for several of his books on algebraic geometry, notable for their informal presentations: * ''Principles of Algebraic Geometry'' , with Phillip Griffiths * ''Geometry of Algebraic Curves, Vol. 1'' , with Enrico Arbarello, Maurizio Cornalba, and Phillip Griffiths * , wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phillip Griffiths
Phillip Augustus Griffiths IV (born October 18, 1938) is an American mathematician, known for his work in the field of geometry, and in particular for the complex manifold approach to algebraic geometry. He was a major developer in particular of the theory of variation of Hodge structure in Hodge theory and moduli theory. He also worked on partial differential equations, coauthored with Shiing-Shen Chern, Robert Bryant and Robert Gardner on Exterior Differential Systems. Professional career He received his BS from Wake Forest College in 1959 and his PhD from Princeton University in 1962 after completing a doctoral dissertation, titled "On certain homogeneous complex manifolds", under the supervision of Donald Spencer. Afterwards, he held positions at University of California, Berkeley (1962–1967) and Princeton University (1967–1972). Griffiths was a professor of mathematics at Harvard University from 1972 to 1983. He was then a Provost and James B. Duke Professor o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a ''positive scalar''. In this context, a convex cone is a cone that is closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets. In this article, only the case of scalars in an ordered field is considered. Definition A subset ''C'' of a vector space ''V'' over an ordered field ''F'' is a cone (or sometimes called a linear cone) if for each ''x'' in ''C'' and positive scalar ''α'' in ''F'', the product ''αx'' is in ''C''. Note that some authors define co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a ''positive scalar''. In this context, a convex cone is a cone that is closed under addition, or, equivalently, a subset of a vector space that is closed under linear combinations with positive coefficients. It follows that convex cones are convex sets. In this article, only the case of scalars in an ordered field is considered. Definition A subset ''C'' of a vector space ''V'' over an ordered field ''F'' is a cone (or sometimes called a linear cone) if for each ''x'' in ''C'' and positive scalar ''α'' in ''F'', the product ''αx'' is in ''C''. Note that some authors define co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kodaira Embedding Theorem
In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials. Kunihiko Kodaira's result is that for a compact Kähler manifold ''M'', with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an ''integral'' cohomology class, there is a complex-analytic embedding of ''M'' into complex projective space of some high enough dimension ''N''. The fact that ''M'' embeds as an algebraic variety follows from its compactness by Chow's theorem. A Kähler manifold with a Hodge metric is occasionally called a Hodge manifold (named after W. V. D. Hodge), so Kodaira's results states that Hodge manifolds are projective. The converse that projective manifolds are Hodge manifolds is more elementary and was already known. Kodaira also proved (Kodaira 1963), by recou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
