K3 surface (original) (raw)
In der Mathematik sind K3-Flächen gewisse komplexe Flächen. Ein klassisches Beispiel ist die Lösungsmenge der Gleichung im dreidimensionalen projektiven Raum. Die Bezeichnung „K3-Fläche“ geht auf André Weil zurück, „in honor of Kummer, Kähler, Kodaira, and the beautiful K2 mountain in Kashmir“.
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| dbo:abstract | In der Mathematik sind K3-Flächen gewisse komplexe Flächen. Ein klassisches Beispiel ist die Lösungsmenge der Gleichung im dreidimensionalen projektiven Raum. Die Bezeichnung „K3-Fläche“ geht auf André Weil zurück, „in honor of Kummer, Kähler, Kodaira, and the beautiful K2 mountain in Kashmir“. (de) In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface in complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to curves or abelian varieties, and yet where a substantial understanding is possible. A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth 4-manifolds. K3 surfaces have been applied to Kac–Moody algebras, mirror symmetry and string theory. It can be useful to think of complex algebraic K3 surfaces as part of the broader family of complex analytic K3 surfaces. Many other types of algebraic varieties do not have such non-algebraic deformations. (en) En géométrie différentielle ou algébrique, les surfaces K3 sont les variétés de Calabi-Yau de plus petite dimension différentes des tores. Ce sont des variétés complexes de dimension complexe 2 compactes et kählériennes. Les surfaces K3 possèdent en outre la propriété d'être les seules variétés de Calabi-Yau distincte du 4-tore T4 d'un point de vue topologique ou différentiel. Cependant, en tant que variété complexe, il y a un nombre infini de surfaces K3 non isomorphes. On peut notamment les distinguer par le biais du (en). André Weil (1958) les nomma en l'honneur des trois géomètres algébristes Kummer, Kähler et Kodaira, et de la montagne K2 au Karakoram. (fr) 数学において、K3曲面 (英: K3 surface) とは、不正則数が 0 で、自明な標準バンドルを持っているという複素解析的、もしくは代数的な滑らかな最小完備曲面をいう。 エンリケス・小平の曲面の分類では、それらは小平次元がゼロの曲面の 4つのクラスのうちの一つである。 K3曲面は、とともに 2次元のカラビ・ヤウ多様体である。ほとんどの複素K3曲面は代数的ではない。このことは、K3曲面を多項式により定義される曲面として射影空間へ埋め込むことができないことを意味する。K3曲面はラマヌジャンが1910年代に発見したが未発表に終わり、後に が再発見して、3人の代数幾何学者（クンマー、ケーラー、小平邦彦）と当時未踏峰だったK2に因みK3曲面と名付けた。 (ja) 대수기하학과 미분기하학에서 K3 곡면(K3曲面, 영어: K3 surface)은 원환면이 아닌 2차원 칼라비-야우 다양체이다. (ko) Een K3-oppervlak is in de algebraïsche meetkunde en de differentiaalmeetkunde, beide deelgebieden van de wiskunde, een complex of algebraïsch, glad, volledig minimaaloppervlak dat is en een triviale kanonieke bundel heeft. Het K3-oppervlak heeft zijn naam te danken aan André Weil, ter ere van de drie meetkundigen Ernst Kummer, Erich Kähler en Kunihiko Kodaira. In de Enriques-Kodaira-classificatie van oppervlakken vormen zij een van de 5 klassen van oppervlakken met Kodaira-dimensie 0. (nl) K3-поверхность есть связная односвязная компактная (то есть комплексное многообразие комплексной размерности два), допускающая нигде не вырожденную голоморфную дифференциальную форму степени два. В алгебраической геометрии, где рассматриваются многообразия над полями иными, нежели комплексные числа, K3-поверхностью называется алгебраическая поверхность с тривиальным , не допускающая алгебраических 1-форм. (ru) A superfície de Kummer se refere à contribuição dada pelo matemático alemão para a geometria. Foi muito mais tarde encontrada por Eddington relacionada com a teoria de Dirac sobre o elétron. (pt) 在數學領域的代數幾何及複流形理論中，K3曲面是一類重要的緊複曲面，在此「曲面」係指複二維，視作實流形則為四維。 K3曲面與二維構成二維的卡拉比-丘流形。複幾何所探討的K3曲面通常不是代數曲面；然而這類曲面首先出現於代數幾何，並以恩斯特·庫默爾、與小平邦彥三位姓氏縮寫為 K 的代數幾何學家命名，也與1950年代被命名的K2峰相映成趣。 (zh) |
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| dbo:wikiPageExternalLink | http://mathematica.sns.it/media/volumi/477/Opere%20matematiche%20I.pdf http://web.math.ucsb.edu/~drm/manuscripts/cortona.pdf http://www.grdb.co.uk/forms/k3 https://magma.maths.usyd.edu.au/magma/handbook/text/1407 http://mathematica.sns.it/opere/479/ http://www.math.polytechnique.fr/~voisin/Articlesweb/Exp.981.C.Voisin.pdf https://www.math.uni-bonn.de/people/huybrech/K3Global.pdf http://projecteuclid.org/euclid.em/1175789798 http://www.numdam.org/item%3Fid=ASENS_1975_4_8_2_235_0 http://www.numdam.org/item%3Fid=SB_1982-1983__25__217_0 http://www.numdam.org/item/AST_1985__126_/ |
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| dbp:first | A.N. (en) |
| dbp:id | k/k055040 (en) |
| dbp:last | Rudakov (en) |
| dbp:quote | Dans la seconde partie de mon rapport, il s'agit des variétés kählériennes dites K3, ainsi nommées en l'honneur de Kummer, Kähler, Kodaira et de la belle montagne K2 au Cachemire. (en) In the second part of my report, we deal with the Kähler varieties known as K3, named in honor of Kummer, Kähler, Kodaira and of the beautiful mountain K2 in Kashmir. (en) |
| dbp:source | André , describing the reason for the name "K3 surface" (en) |
| dbp:title | K3 surface (en) |
| dbp:width | 30.0 |
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| gold:hypernym | dbr:Surface |
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| rdfs:comment | In der Mathematik sind K3-Flächen gewisse komplexe Flächen. Ein klassisches Beispiel ist die Lösungsmenge der Gleichung im dreidimensionalen projektiven Raum. Die Bezeichnung „K3-Fläche“ geht auf André Weil zurück, „in honor of Kummer, Kähler, Kodaira, and the beautiful K2 mountain in Kashmir“. (de) 数学において、K3曲面 (英: K3 surface) とは、不正則数が 0 で、自明な標準バンドルを持っているという複素解析的、もしくは代数的な滑らかな最小完備曲面をいう。 エンリケス・小平の曲面の分類では、それらは小平次元がゼロの曲面の 4つのクラスのうちの一つである。 K3曲面は、とともに 2次元のカラビ・ヤウ多様体である。ほとんどの複素K3曲面は代数的ではない。このことは、K3曲面を多項式により定義される曲面として射影空間へ埋め込むことができないことを意味する。K3曲面はラマヌジャンが1910年代に発見したが未発表に終わり、後に が再発見して、3人の代数幾何学者（クンマー、ケーラー、小平邦彦）と当時未踏峰だったK2に因みK3曲面と名付けた。 (ja) 대수기하학과 미분기하학에서 K3 곡면(K3曲面, 영어: K3 surface)은 원환면이 아닌 2차원 칼라비-야우 다양체이다. (ko) Een K3-oppervlak is in de algebraïsche meetkunde en de differentiaalmeetkunde, beide deelgebieden van de wiskunde, een complex of algebraïsch, glad, volledig minimaaloppervlak dat is en een triviale kanonieke bundel heeft. Het K3-oppervlak heeft zijn naam te danken aan André Weil, ter ere van de drie meetkundigen Ernst Kummer, Erich Kähler en Kunihiko Kodaira. In de Enriques-Kodaira-classificatie van oppervlakken vormen zij een van de 5 klassen van oppervlakken met Kodaira-dimensie 0. (nl) K3-поверхность есть связная односвязная компактная (то есть комплексное многообразие комплексной размерности два), допускающая нигде не вырожденную голоморфную дифференциальную форму степени два. В алгебраической геометрии, где рассматриваются многообразия над полями иными, нежели комплексные числа, K3-поверхностью называется алгебраическая поверхность с тривиальным , не допускающая алгебраических 1-форм. (ru) A superfície de Kummer se refere à contribuição dada pelo matemático alemão para a geometria. Foi muito mais tarde encontrada por Eddington relacionada com a teoria de Dirac sobre o elétron. (pt) 在數學領域的代數幾何及複流形理論中，K3曲面是一類重要的緊複曲面，在此「曲面」係指複二維，視作實流形則為四維。 K3曲面與二維構成二維的卡拉比-丘流形。複幾何所探討的K3曲面通常不是代數曲面；然而這類曲面首先出現於代數幾何，並以恩斯特·庫默爾、與小平邦彥三位姓氏縮寫為 K 的代數幾何學家命名，也與1950年代被命名的K2峰相映成趣。 (zh) In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface in complex projective 3-space. (en) En géométrie différentielle ou algébrique, les surfaces K3 sont les variétés de Calabi-Yau de plus petite dimension différentes des tores. Ce sont des variétés complexes de dimension complexe 2 compactes et kählériennes. Les surfaces K3 possèdent en outre la propriété d'être les seules variétés de Calabi-Yau distincte du 4-tore T4 d'un point de vue topologique ou différentiel. Cependant, en tant que variété complexe, il y a un nombre infini de surfaces K3 non isomorphes. On peut notamment les distinguer par le biais du (en). (fr) |
| rdfs:label | K3-Fläche (de) K3 (géométrie) (fr) K3 surface (en) K3 곡면 (ko) K3-oppervlak (nl) K3曲面 (ja) K3 (geometria) (pt) K3-поверхность (ru) K3曲面 (zh) |
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