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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Logarithmic form ： ウィキペディア英語版 Logarithmic form In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. Let ''X'' be a complex manifold, and ''D'' ⊂ ''X'' a divisor and ω a holomorphic ''p''-form on ''X''−''D''. If ω and ''d''ω have a pole of order at most one along ''D'', then ω is said to have a logarithmic pole along ''D''. ω is also known as a logarithmic ''p''-form. The logarithmic ''p''-forms make up a subsheaf of the meromorphic ''p''-forms on ''X'' with a pole along ''D'', denoted :$\backslash Omega^p\_X(\backslash log\; D).$ In the theory of Riemann surfaces, one encounters logarithmic one-forms which have the local expression :$\backslash omega\; =\; \backslash frac\; =\backslash left(\backslash frac\; +\; \backslash frac\backslash right)dz$ for some meromorphic function (resp. rational function) $f(z)\; =\; z^mg(z)$, where ''g'' is holomorphic and non-vanishing at 0, and ''m'' is the order of ''f'' at ''0''. That is, for some open covering, there are local representations of this differential form as a logarithmic derivative (modified slightly with the exterior derivative ''d'' in place of the usual differential operator ''d/dz''). Observe that ω has only simple poles with integer residues. On higher-dimensional complex manifolds, the Poincaré residue is used to describe the distinctive behavior of logarithmic forms along poles. ==Holomorphic log complex== By definition of $\backslash Omega^p\_X(\backslash log\; D)$ and the fact that exterior differentiation ''d'' satisfies ''d''2 = 0, one has :$d\backslash Omega^p\_X(\backslash log\; D)(U)\backslash subset\; \backslash Omega^\_X(\backslash log\; D)(U)$. This implies that there is a complex of sheaves $(\; \backslash Omega^\_X(\backslash log\; D),\; d)$, known as the ''holomorphic log complex'' corresponding to the divisor ''D''. This is a subcomplex of $j\_$ *\Omega^_ , where $j:X-D\backslash rightarrow\; X$ is the inclusion and $\backslash Omega^\_$ is the complex of sheaves of holomorphic forms on ''X''−''D''. Of special interest is the case where ''D'' has simple normal crossings. Then if $\backslash$ are the smooth, irreducible components of ''D'', one has $D\; =\; \backslash sum\; D\_$ with the $D\_$ meeting transversely. Locally ''D'' is the union of hyperplanes, with local defining equations of the form $z\_1\backslash cdots\; z\_k\; =\; 0$ in some holomorphic coordinates. One can show that the stalk of $\backslash Omega^1\_X(\backslash log\; D)$ at ''p'' satisfies〔Chris A.M. Peters; Joseph H.M. Steenbrink (2007). Mixed Hodge Structures. Springer. ISBN 978-3-540-77017-6〕 :$\backslash Omega\_X^1(\backslash log\; D)\_p\; =\; \backslash mathcal\_\backslash frac\backslash oplus\backslash cdots\backslash oplus\backslash mathcal\_\backslash frac\; \backslash oplus\; \backslash mathcal\_dz\_\; \backslash oplus\; \backslash cdots\; \backslash oplus\; \backslash mathcal\_dz\_n$ and that :$\backslash Omega\_X^k(\backslash log\; D)\_p\; =\; \backslash bigwedge^k\_\; \backslash Omega\_X^1(\backslash log\; D)\_p$. Some authors, e.g.,〔Phillip A. Griffiths; Joseph Harris (1979). Principles of Algebraic Geometry. Wiley-Interscience. ISBN 0-471-05059-8.〕 use the term ''log complex'' to refer to the holomorphic log complex corresponding to a divisor with normal crossings. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Logarithmic form」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.