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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 Kodaira dimension ： ウィキペディア英語版 Kodaira dimension In algebraic geometry, the Kodaira dimension κ(''X'') (or canonical dimension) measures the size of the canonical model of a projective variety ''X''. Igor Shafarevich introduced an important numerical invariant of surfaces with the notation κ in the seminar Shafarevich 1965. extended it and defined the Kodaira dimension for higher dimensional varieties (under the name of canonical dimension), and later named it after Kunihiko Kodaira in . ==The plurigenera== The canonical bundle of a smooth algebraic variety ''X'' of dimension ''n'' over a field is the line bundle of ''n''-forms, :$\backslash ,\backslash !K\_X\; =\; \backslash bigwedge^n\backslash Omega^1\_X,$ which is the ''n''th exterior power of the cotangent bundle of ''X''. For an integer ''d'', the ''d''th tensor power of ''KX'' is again a line bundle. For ''d ≥ 0'', the vector space of global sections ''H0(X,KXd)'' has the remarkable property that it is a birational invariant of smooth projective varieties ''X''. That is, this vector space is canonically identified with the corresponding space for any smooth projective variety which is isomorphic to ''X'' outside lower-dimensional subsets. For ''d ≥ 0'', the ''d''th plurigenus of ''X'' is defined as the dimension of the vector space of global sections of ''KXd'': :$P\_d\; =\; h^0(X,\; K\_X^d)\; =\; \backslash operatorname\backslash \; H^0(X,\; K\_X^d).$ The plurigenera are important birational invariants of an algebraic variety. In particular, the simplest way to prove that a variety is not rational (that is, not birational to projective space) is to show that some plurigenus ''Pd'' with ''d > 0'' is not zero. If the space of sections of ''KXd'' is nonzero, then there is a natural rational map from ''X'' to the projective space :$\backslash mathbf(H^0(X,\; K\_X^d))\; =\; \backslash mathbf^$, called the ''d''-canonical map. The canonical ring ''R(KX)'' of a variety ''X'' is the graded ring :$R(K\_X)\; :=\backslash bigoplus\_\; H^0(X,K\_X^d).$ Also see geometric genus and arithmetic genus. The Kodaira dimension of ''X'' is defined to be −∞ if the plurigenera ''Pd'' are zero for all ''d'' > 0; otherwise, it is the minimum κ such that ''Pd/dκ'' is bounded. The Kodaira dimension of an ''n''-dimensional variety is either −∞ or an integer in the range from 0 to ''n''. 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「Kodaira dimension」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.