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| In [[algebraic geometry]], the '''Kodaira dimension''' κ(''X'') measures the size of the [[canonical ring|canonical model]] of a [[projective variety]] ''X''.
| | == who died still hard to say.' Luo Feng eyes firmly. == |
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| Kodaira dimension is named for [[Kunihiko Kodaira]]. The name and the notation κ were introduced by [[Igor Shafarevich]] in the seminar [[#refShafarevich1965|Shafarevich 1965]].
| | , And I do not have to intervene. '<br><br>Hung, is superior, what identity!<br><br>Warrior thing, if not necessary, he really bother to intervene.<br><br>'That vulture Li Yao to the continent of Australia,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_37.htm オークリー野球サングラス], who died still hard to say.' Luo Feng eyes firmly.<br><br>total flood the main hall nodded, 'Luo Feng,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_44.htm オークリー サングラス 野球], you are my ultimate martial arts person,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_44.htm オークリー サングラス 野球], I remind you that - it was once the ancient civilization ruins vulture Yao-out success, to get a 'black god' this dark god. Set main thing is that you can form a protective armor meaning heart, and even the whole body can be hundred percent coverage, including any location, such as the nose and mouth. '<br><br>Luo Feng hesitated,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_45.htm オークリー サングラス ジョウボーン].<br><br>'Once hundred percent coverage,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_62.htm オークリー サングラス 交換レンズ], your knife, there is no shot in black suits of God.' total flood the main hall looked Luo Feng his pocket, 'of course, did not always connected to all parts of the nose and mouth and other co- time to cover, especially the eyes, the eyes of a cover,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_36.htm オークリー サングラス レーダー], not out of sight? '<br><br>Luo Feng jing ,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_9.htm サングラス オークリー 偏光]......<br><br>the |
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| | <li>[http://www.fooyso.com/zhidao/question.php?qid=399710 http://www.fooyso.com/zhidao/question.php?qid=399710]</li> |
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| | </ul> |
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| ==The plurigenera== | | == Huaxia Guo == |
| The [[canonical bundle]] of a smooth [[algebraic variety]] ''X'' of dimension ''n'' over a field is the [[line bundle]] of ''n''-forms,
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| <!-- Using \,\! to force PNG rendering, else formula won't show up (used again below) -->
| | Star strong run for,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_30.htm オークリー サングラス フロッグスキン], and a king-level monster killing, perhaps very hard and dangerous,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_75.htm オークリー サングラス 調整], and sometimes had to escape.<br>Luo Feng<br>can not the same!<br><br>him to run for that one result - victory!<br><br>from day dawn until three in the afternoon, Luo Feng died in the hands of the king of the monster level reached frightening 37! Gee, Huaxia Guo other strong all-star line add up, do not kill more than half of the Luo Feng!<br><br>reputation earthquake,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_3.htm オークリーサングラス ランキング]!<br><br>a time, Huaxia Guo, even these video broadcast in other countries,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_11.htm オークリー サングラス レディース], many people are starting Luofeng worship.<br><br>......<br><br>Yet,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_68.htm ゴルフ オークリー サングラス]!<br><br>killed 37 monsters king class of humans is a great encouragement. But for the purposes of the sea monster party ...... they are extremely angry! Powerful monsters wise,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_77.htm オークリー サングラス 通販], was slain king level 37 monsters,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_60.htm ロードバイク サングラス オークリー], soon provoked one of the monsters in the waters of two royal.<br><br>Octopussy imperial beast,[http://www.alleganycountyfair.org/_vti_cnf/rakuten_oakley_60.htm ロードバイク サングラス オークリー]!<br><br>this beast Octopussy |
| :<math>\,\!K_X = \bigwedge^n\Omega_X,</math> | | 相关的主题文章： |
| | | <ul> |
| which is the ''n''th [[exterior power]] of the [[cotangent bundle]] of ''X''.
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| For an integer ''d'', the ''d''th tensor power of ''K<sub>X</sub>'' is again a line bundle.
| | <li>[http://bbs.chinaacc.tv/home.php?mod=space&uid=186363 http://bbs.chinaacc.tv/home.php?mod=space&uid=186363]</li> |
| For ''d ≥ 0'', the vector space of global sections ''H<sup>0</sup>(X,K<sub>X</sub><sup>d</sup>)'' has the remarkable property that it is a [[birational geometry|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.
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| | | <li>[http://www.sandhillberries.com/cgi-bin/active/guestbook.cgi http://www.sandhillberries.com/cgi-bin/active/guestbook.cgi]</li> |
| For ''d ≥ 0'', the
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| ''d''th '''plurigenus''' of ''X'' is defined as the dimension of the vector space
| | <li>[http://www.kindyroo.com/bbs/forum.php?mod=viewthread&tid=231914&fromuid=82275 http://www.kindyroo.com/bbs/forum.php?mod=viewthread&tid=231914&fromuid=82275]</li> |
| of global sections of ''K<sub>X</sub><sup>d</sub>'':
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| | | </ul> |
| :<math>P_d = h^0(X, K_X^d) = \operatorname{dim}\ H^0(X, K_X^d).</math>
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| 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 ''P<sub>d</sub>'' with ''d > 0''
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| is not zero. If the space of sections of ''K<sub>X</sub><sup>d</sup>'' is nonzero, then there is a natural rational map from ''X'' to the projective space
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| :<math>\mathbf{P}(H^0(X, K_X^d)) = \mathbf{P}^{P_d - 1}</math>,
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| called the ''d''-'''canonical map'''. The [[canonical ring]] ''R(K<sub>X</sub>)'' of a variety ''X'' is the graded ring
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| :<math> R(K_X) :=\bigoplus_{d\geq 0} H^0(X,K_X^d). </math>
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| Also see [[geometric genus]] and [[arithmetic genus]].
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| The '''Kodaira dimension''' of ''X'' is defined to be −∞ if the plurigenera ''P<sub>d</sub>'' are zero for all ''d'' > 0; otherwise, it is the minimum κ such that ''P<sub>d</sub>/d<sup>κ</sup>'' is bounded. The Kodaira dimension of an ''n''-dimensional variety is either −∞ or an integer in the range from 0 to ''n''.
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| ===Interpretations of the Kodaira dimension===
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| The following integers are equal. A good reference is {{harvtxt|Lazarsfeld|2004}}, Theorem 2.1.33.
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| * The dimension of the [[Proj construction]] Proj ''R(K<sub>X</sub>)'' (this variety is called the '''canonical model''' of ''X''; it only depends on the birational equivalence class of ''X'').
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| * The dimension of the image of the ''d''-canonical mapping for all positive multiples ''d'' of some positive integer ''d''<sub>0</sub>.
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| * The [[transcendence degree]] of ''R'', minus one, i.e. ''t'' − 1, where ''t'' is the number of [[algebraically independent]] generators one can find.
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| * The rate of growth of the plurigenera: that is, the smallest number κ such that ''P<sub>d</sub>/d<sup>κ</sup>'' is bounded. In [[Big O notation]], it is the minimal κ such that ''P<sub>d</sub> = O(d<sup>κ</sup>)''.
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| When the plurigenera ''P<sub>d</sub>'' are zero for all positive ''d'' (so the canonical ring ''R(K<sub>X</sub>)'' is equal to the base field ''R<sub>0</sub>''), we have to define the Kodaira dimension to be −∞ rather than −1, in order to make the formula ''κ(X × Y) = κ(X) + κ(Y)'' true in all cases. For example, the Kodaira dimension of '''P'''<sup>1</sup> × ''X'' is −∞ for all varieties ''X''. That convention is also essential in the statement of the [[Kodaira_dimension#Application_to_classification|Iitaka conjecture]].
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| ===Application===
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| The Kodaira dimension gives a useful rough division of all algebraic varieties into several classes.
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| Varieties with low Kodaira dimension can be considered special, while varieties of maximal Kodaira dimension are said to be of [[#General type|general type]].
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| Geometrically, there is a very rough correspondence between Kodaira dimension and curvature: negative Kodaira dimension corresponds to positive curvature, zero Kodaira dimension corresponds to flatness, and maximum Kodaira dimension (general type) corresponds to negative curvature.
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| The specialness of varieties of low Kodaira dimension is analogous to the specialness of Riemannian manifolds of positive curvature (and general type corresponds to the genericity of non-positive curvature); see [[Riemannian_geometry#Local_to_global_theorems|classical theorems]], especially on ''Pinched sectional curvature'' and ''Positive curvature''.
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| These statements are made more precise below.
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| ===Dimension 1===
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| Smooth projective curves are discretely classified by [[genus (mathematics)|genus]], which can be any [[natural number]] ''g'' = 0, 1, ....
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| By "discretely classified", we mean that for a given genus, there is a connected, irreducible [[moduli space]] of curves of that genus.
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| The Kodaira dimension of a curve ''X'' is:
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| * κ = −∞: genus 0 (the [[projective line]] '''P'''<sup>1</sup>): ''K<sub>X</sub>'' is not effective, ''P<sub>d</sub> = 0'' for all ''d > 0''.
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| * κ = 0: genus 1 ([[elliptic curve]]s): ''K<sub>X</sub>'' is a [[trivial bundle]], ''P<sub>d</sub> = 1'' for all ''d ≥ 0''.
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| * κ = 1: genus ''g ≥ 2'': ''K<sub>X</sub>'' is [[ample line bundle|ample]], ''P<sub>d</sub>=(2d−1)(g−1)'' for all ''d ≥ 2''.
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| Compare with the [[Uniformization theorem]] for surfaces (real surfaces, since a complex curve
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| has real dimension 2): Kodaira dimension −∞ corresponds to positive curvature, Kodaira dimension 0 corresponds to flatness, Kodaira dimension 1 corresponds to negative curvature. Note that most algebraic curves are of general type: in the moduli space of curves, two connected components correspond to curves not of general type, while all the other components correspond to curves of general type. Further, the space of curves of genus 0 is a point, the space of curves of genus 1 has (complex) dimension 1, and the space of curves of genus ''g'' ≥ 2 has dimension 3''g''−3.
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| :{| class="wikitable"
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| ! colspan="3"| the classification table of algebraic curves
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| |-
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| ! rowspan="2"| Kodaira dimension <br /> κ(C)
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| |-
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| ! [[genus]] of C : g(C)
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| ! structure
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| |-
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| ! <math>1</math>
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| | <math>\ge 2</math>
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| | curve of [[#general type|general type]]
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| |-
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| ! <math>0</math>
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| | <math>1</math>
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| | [[elliptic curve]]
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| |-
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| ! <math>-\infty</math>
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| | <math>0</math>
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| | the [[projective space|projective line]] <math>\mathbb{P}^1</math>
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| |-
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| |}
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| ===Dimension 2===
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| The [[Enriques-Kodaira classification]] classifies algebraic surfaces: coarsely by Kodaira dimension, then in more detail within a given Kodaira dimension. To give some simple examples: the product '''P'''<sup>1</sup> × ''X'' has Kodaira dimension −∞ for any curve ''X''; the product of two curves of genus 1 (an abelian surface) has Kodaira dimension 0; the product of a curve of genus 1 with a curve of genus at least 2 (an elliptic surface) has Kodaira dimension 1; and the product of two curves of genus at least 2 has Kodaira dimension 2 and hence is of [[#General type|general type]].
| |
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| :{| class="wikitable"
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| ! colspan="4"| the classification table of algebraic surfaces
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| |-
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| ! rowspan="2"| Kodaira dimension <br /> κ(C)
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| |-
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| ! [[geometric genus]] <br /> p<sub>g</sub>
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| ! [[irregularity of a surface|irregularity]] <br />q
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| ! structure
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| |-
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| ! <math>2</math>
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| | surface of [[#general type|general type]]
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| |-
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| ! <math>1</math>
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| | [[elliptic surface]]
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| |-
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| ! rowspan="4"| <math>0</math> | |
| | <math>1</math>
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| | <math>2</math>
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| | [[abelian surface]]
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| |-
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| | <math>0</math>
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| | <math>1</math>
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| | [[hyperelliptic surface]]
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| |-
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| | <math>1</math>
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| | <math>0</math>
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| | [[K3 surface]]
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| |-
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| | <math>0</math>
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| | <math>0</math>
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| | [[Enriques surface]]
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| |-
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| ! rowspan="2"| <math>-\infty</math>
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| | <math>0</math>
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| | <math>\ge1</math>
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| | [[ruled surface]]
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| |-
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| | <math>0</math>
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| | <math>0</math>
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| | [[rational surface]]
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| |-
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| |}
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| For a surface ''X'' of general type, the image of the ''d''-canonical map is birational to ''X'' if ''d'' ≥ 5.
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| ===Any dimension=== | |
| Rational varieties (varieties birational to projective space) have Kodaira dimension −∞. [[Abelian variety|Abelian varieties]] and [[Calabi-Yau]] manifolds (in dimension 1, [[elliptic curve]]s; in dimension 2, [[complex tori]] and [[K3 surface]]s) have Kodaira dimension zero (corresponding to admitting flat metrics and Ricci flat metrics, respectively).
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| Any variety covered by [[rational curve]]s (nonconstant maps from '''P'''<sup>1</sup>),
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| called a uniruled variety, has Kodaira dimension −∞. Conversely, the main conjectures of [[minimal model program|minimal model theory]] (notably the abundance conjecture) would imply that every variety of Kodaira dimension −∞ is uniruled. This converse is known for varieties of dimension at most 3.
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| {{harvtxt|Siu|2002}} proved the invariance of plurigenera under deformations for all smooth complex projective varieties. In particular, the Kodaira dimension does not change when the complex structure of the manifold is changed continuously.
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| :{| class="wikitable"
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| ! colspan="4"| the classification table of algebraic three-folds
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| |-
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| ! rowspan="2"| Kodaira dimension <br /> κ(C)
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| |-
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| ! [[geometric genus]] <br /> p<sub>g</sub>
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| ! [[irregularity of a surface|irregularity]] <br />q
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| ! examples
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| |-
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| ! <math>3</math>
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| | three-fold of [[#general type|general type]]
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| |-
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| ! <math>2</math>
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| | fibration over a surface with general fiber an [[elliptic curve]]
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| |-
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| ! <math>1</math>
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| | fibration over a curve with general fiber a surface with κ = 0
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| |-
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| ! rowspan="4"| <math>0</math>
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| | <math>1</math>
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| | <math>3</math>
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| | [[abelian variety]]
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| |-
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| | <math>0</math>
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| | <math>2</math>
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| | [[fiber bundle]] over an abelian surface whose fibers are elliptic curves
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| |-
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| | <math>0</math> or <math>1</math>
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| | <math>1</math>
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| | [[fiber bundle]] over an elliptic curve whose fibers are surfaces with κ = 0
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| |-
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| | <math>0</math> or <math>1</math>
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| | <math>0</math>
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| | [[Calabi-Yau manifold|Calabi-Yau]] 3-fold
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| |-
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| ! rowspan="2"| <math>-\infty</math>
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| | <math>0</math>
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| | <math>\ge1</math>
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| | [[ruled variety|uniruled]] 3-folds
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| |-
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| | <math>0</math>
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| | <math>0</math>
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| | [[rational variety|rational]] 3-folds, [[Fano variety|Fano]] 3-folds, and others
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| |-
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| |}
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| A '''fibration''' of normal projective varieties ''X'' → ''Y'' means a surjective morphism with connected fibers.
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| For a 3-fold ''X'' of general type, the image of the ''d''-canonical map is birational to ''X'' if ''d'' ≥ 61.
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| ==General type== | |
| A variety of '''general type''' ''X'' is one of maximal Kodaira dimension (Kodaira dimension equal to its dimension):
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| :<math>\kappa(X) = \operatorname{dim}\ X.</math> | |
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| Equivalent conditions are that the line bundle ''K<sub>X</sub>'' is [[big line bundle|big]], or that the ''d''-canonical map is generically injective (that is, a birational map to its image) for ''d'' sufficiently large.
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| For example, a variety with [[ample line bundle|ample]] canonical bundle is of general type.
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| In some sense, most algebraic varieties are of general type. For example, a smooth hypersurface of degree ''d'' in the ''n''-dimensional projective space is of general type if and only if ''d > n+1''. So we can say that most smooth hypersurfaces in projective space are of general type.
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| Varieties of general type seem too complicated to classify explicitly, even for surfaces. Nonetheless, there are some strong positive results about varieties of general type. For example, Bombieri showed in 1973 that the ''d''-canonical map of any complex surface of general type is birational for every ''d ≥ 5''. More generally, Hacon-McKernan, Takayama, and Tsuji showed in 2006 that for every positive integer ''n'', there is a constant ''c(n)'' such that the ''d''-canonical map of any complex ''n''-dimensional variety of general type is birational when ''d ≥ c(n)''.
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| Furthermore, varieties of general type have a only finite group as their automorphism.
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| ==Application to classification== | |
| The '''Iitaka conjecture''' states that the Kodaira dimension of a fibration is at least the sum of the Kodaira dimension of the base and the Kodaira dimension of a general fiber; see {{harvtxt|Mori|1987}} for a survey. The Iitaka conjecture helped to inspire the development of [[minimal model program|minimal model theory]] in the 1970s and 1980s. It is now known in many cases, and would follow in general from the main conjectures of minimal model theory, notably the abundance conjecture.
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| Minimal model theory also predicts that every algebraic variety is either uniruled (covered by rational curves) or birational to a family of varieties over the canonical model, with general fiber a [[Calabi-Yau]] variety. To some extent, this would reduce the study of arbitrary varieties to the cases of Calabi-Yau varieties and varieties of general type.
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| ==The relationship to Moishezon manifolds==
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| Nakamura and Ueno proved the following additivity formula for complex manifolds ({{harvtxt|Ueno|1975}}). Although the base space is not required to be algebraic, the assumption that all the fibers are isomorphic is very special. Even with this assumption, the formula can fail when the fiber is not Moishezon.
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| :Let π: V → W be an analytic fiber bundle of compact complex manifolds, meaning that π is locally a product (and so all fibers are isomorphic as complex manifolds). Suppose that the fiber F is a [[Moishezon manifold]]. Then
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| :<math>\kappa(V)=\kappa(F)+\kappa(W).</math>
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| ==See also==
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| *[[Birational geometry]]
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| *[[Enriques-Kodaira classification]]
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| *[[Iitaka dimension]]
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| *[[Minimal model program]]
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| *[[Moishezon manifold]]
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| ==References==
| |
| *{{eom|id=Kodaira_dimension|authorlink=Igor Dolgachev|last=Dolgachev|first=I, |title=Kodaira_dimension}}
| |
| *{{Citation | last1=Lazarsfeld | first1=Robert | mr=2095471 | title=Positivity in algebraic geometry | volume=1 | publisher=Springer-Verlag | location=Berlin | year=2004 | ISBN=3-540-22533-1}}
| |
| *{{Citation | last1=Mori | first1=Shigefumi | mr =0927961 | title=Algebraic geometry (Bowdoin, 1985) | chapter=Classification of higher-dimensional varieties | pages = 269–331 | series = Proceedings of Symposia in Pure Mathematics | volume = 46, Part 1 | publisher=American Mathematical Society | year=1987}}
| |
| *{{Citation | ref=refShafarevich1965 | last1=Shafarevich | first1=Igor R. | last2=Averbuh | first2=B. G. | last3=Vaĭnberg | first3=Ju. R. | last4=Zhizhchenko | first4=A. B. | last5=Manin | first5=Ju. I. | last6=Moĭshezon | first6=B. G. | last7=Tjurina | first7=G. N. | last8=Tjurin | first8=A. N. | title=Algebraic surfaces | id={{MathSciNet | id = 0190143}} | year=1965 | journal=Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova | issn=0371-9685 | volume=75 | pages=1–215}}
| |
| *{{Citation | last1=Siu | first1 = Y.-T. | title=Complex geometry (Gottingen, 2000) | chapter=Extension of twisted pluricanonical sections with plurisubharmonic weight and invariance of semi-positively twisted plurigenera for manifolds not necessarily of general type | mr=1922108 | year=2002 | pages = 223–277 | publisher=[[Springer-Verlag]] | location=Berlin}}
| |
| *{{Citation | last1=Ueno | first1 = Kenji | mr=0506253 | title=Classification theory of algebraic varieties and compact complex spaces | year=1975 | series=Lecture Notes in Mathematics | volume=439 | publisher=[[Springer-Verlag]] }}
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| [[Category:Birational geometry]]
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| [[Category:Dimension]]
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who died still hard to say.' Luo Feng eyes firmly.
, And I do not have to intervene. '
Hung, is superior, what identity!
Warrior thing, if not necessary, he really bother to intervene.
'That vulture Li Yao to the continent of Australia,オークリー野球サングラス, who died still hard to say.' Luo Feng eyes firmly.
total flood the main hall nodded, 'Luo Feng,オークリー サングラス 野球, you are my ultimate martial arts person,オークリー サングラス 野球, I remind you that - it was once the ancient civilization ruins vulture Yao-out success, to get a 'black god' this dark god. Set main thing is that you can form a protective armor meaning heart, and even the whole body can be hundred percent coverage, including any location, such as the nose and mouth. '
Luo Feng hesitated,オークリー サングラス ジョウボーン.
'Once hundred percent coverage,オークリー サングラス 交換レンズ, your knife, there is no shot in black suits of God.' total flood the main hall looked Luo Feng his pocket, 'of course, did not always connected to all parts of the nose and mouth and other co- time to cover, especially the eyes, the eyes of a cover,オークリー サングラス レーダー, not out of sight? '
Luo Feng jing ,サングラス オークリー 偏光......
the
相关的主题文章：
Huaxia Guo
Star strong run for,オークリー サングラス フロッグスキン, and a king-level monster killing, perhaps very hard and dangerous,オークリー サングラス 調整, and sometimes had to escape.
Luo Feng
can not the same!
him to run for that one result - victory!
from day dawn until three in the afternoon, Luo Feng died in the hands of the king of the monster level reached frightening 37! Gee, Huaxia Guo other strong all-star line add up, do not kill more than half of the Luo Feng!
reputation earthquake,オークリーサングラス ランキング!
a time, Huaxia Guo, even these video broadcast in other countries,オークリー サングラス レディース, many people are starting Luofeng worship.
......
Yet,ゴルフ オークリー サングラス!
killed 37 monsters king class of humans is a great encouragement. But for the purposes of the sea monster party ...... they are extremely angry! Powerful monsters wise,オークリー サングラス 通販, was slain king level 37 monsters,ロードバイク サングラス オークリー, soon provoked one of the monsters in the waters of two royal.
Octopussy imperial beast,ロードバイク サングラス オークリー!
this beast Octopussy
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