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Bott Residue
In mathematics, the Bott residue formula, introduced by , describes a sum over the fixed points of a holomorphic vector field of a compact complex manifold. Statement If ''v'' is a holomorphic vector field on a compact complex manifold ''M'', then : \sum_\frac = \int_M P(i\Theta/2\pi) where *The sum is over the fixed points ''p'' of the vector field ''v'' *The linear transformation ''A''''p'' is the action induced by ''v'' on the holomorphic tangent space at ''p'' *''P'' is an invariant polynomial function of matrices of degree dim(''M'') *Θ is a curvature matrix of the holomorphic tangent bundle See also *Atiyah–Bott fixed-point theorem *Holomorphic Lefschetz fixed-point formula References * *{{Citation , last1=Griffiths , first1=Phillip , author1-link=Phillip Griffiths , last2=Harris , first2=Joseph , author2-link=Joe Harris (mathematician) , title=Principles of algebraic geometry , publisher=John Wiley & Sons John Wiley & Sons, Inc., commonly known as Wiley ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Point (mathematics)
A fixed point (sometimes shortened to fixpoint, also known as an invariant point) is a value that does not change under a given transformation. Specifically, in mathematics, a fixed point of a function is an element that is mapped to itself by the function. In physics, the term fixed point can refer to a temperature that can be used as a reproducible reference point, usually defined by a phase change or triple point. Fixed point of a function Formally, is a fixed point of a function if belongs to both the domain and the codomain of , and . For example, if is defined on the real numbers by f(x) = x^2 - 3 x + 4, then 2 is a fixed point of , because . Not all functions have fixed points: for example, , has no fixed points, since is never equal to for any real number. In graphical terms, a fixed point means the point is on the line , or in other words the graph of has a point in common with that line. Fixed-point iteration In numerical analysis, ''fixed-point iter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (''analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''regular fu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Manifold
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] |
Atiyah–Bott Fixed-point Theorem
In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds ''M'', which uses an elliptic complex on ''M''. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem. Formulation The idea is to find the correct replacement for the Lefschetz number, which in the classical result is an integer counting the correct contribution of a fixed point of a smooth mapping f\colon M \to M. Intuitively, the fixed points are the points of intersection of the graph of ''f'' with the diagonal (graph of the identity mapping) in M\times M, and the Lefschetz number thereby becomes an intersection number. The Atiyah–Bott theorem is an equation in which the LHS must be the outcome of a global topological (homological) calcul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Lefschetz Fixed-point Formula
In mathematics, the Holomorphic Lefschetz formula is an analogue for complex manifolds of the Lefschetz fixed-point formula that relates a sum over the fixed point (mathematics), fixed points of a holomorphic vector field of a compact complex manifold to a sum over its Dolbeault cohomology groups. Statement If ''f'' is an automorphism of a compact complex manifold ''M'' with isolated fixed points, then : \sum_\frac = \sum_q(-1)^q\operatorname(f^*, H^_(M)) where *The sum is over the fixed points ''p'' of ''f'' *The linear transformation ''A''''p'' is the action induced by ''f'' on the holomorphic tangent space at ''p'' See also *Bott residue formula References *{{Citation , last1=Griffiths , first1=Phillip , author1-link=Phillip Griffiths , last2=Harris , first2=Joseph , author2-link=Joe Harris (mathematician) , title=Principles of algebraic geometry , publisher=John Wiley & Sons , location=New York , series=Wiley Classics Library , isbn=978-0-471-05059-9 , mr=128 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orange, New Je ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
