About: Fano variety

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In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano , ), is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties.

Attributes	Values
rdf:type	Thing work
rdfs:label	Fano-Varietät (de) Fano variety (en) ファノ多様体 (ja) 파노 다양체 (ko) Fano-variëteit (nl)
rdfs:comment	In der algebraischen Geometrie, einem Teilgebiet der Mathematik, ist eine Fano-Varietät eine vollständige Varietät über einem Körper , deren ist. Eine Fano-Mannigfaltigkeit ist eine singularitäten-freie komplexe Fano-Varietät. (de) In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano , ), is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties. (en) 대수기하학에서 파노 다양체(영어: Fano variety)는 사영 공간과 유사하게, 반표준 인자가 풍부한 인자를 이루는 대수다양체이다. (ko) 代数幾何学では、ファノ多様体(Fano variety)は、( Fano , ) により導入され、多様体上の反標準バンドルが豊富な(complete variety) X のことを言う。この定義は、X がある定義体上で(smooth)なことを前提としているが、極小モデルプログラムでは、端末特異点(canonical singularity)やklt特異点(klt singularity)（川又対数端末特異点）といった、様々なタイプの特異点を持ったファノ多様体の研究も進められていた。 (ja) In algebraïsche meetkunde is een Fano-variëteit, geïntroduceerd door Gino Fano, een niet-singuliere complete variëteit waarvan de anti-kanonieke bundel een is. In het bijzonder hebben alle Fano-variëteiten een Kodaira-dimensie −∞. (nl)
dcterms:subject	3-folds Algebraic geometry
Wikipage page ID	3095929 (xsd:integer)
Wikipage revision ID	1101817985 (xsd:integer)
Link from a Wikipage to another Wikipage	Yoichi Miyaoka Rational variety Algebraic geometry of projective spaces Betti number Del Pezzo surface Kähler–Einstein metric Weighted projective space Gino Fano Calabi conjecture Complete intersection Yuri Manin Adjunction formula K-stability of Fano varieties Algebraic curve Algebraic geometry Ample line bundle 3-folds Curvature Isomorphism Kodaira vanishing theorem Simply connected space Projective line Projective variety Herbert Clemens Hypersurface Algebraic geometry János Kollár Moduli space Phillip Griffiths Kodaira dimension Canonical bundle Canonical singularity Shigefumi Mori Minimal model program Sheaf cohomology Smooth scheme Periodic table of shapes Universal cover Fubini–Study Variety of general type Springer-Verlag First Chern class Myers' theorem Very ample Complete algebraic variety Anticanonical bundle Structure sheaf dbr:Vasily_Iskovskikh
Link from a Wikipage to an external page	https://fanography.pythonanywhere.com/ http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/%3FIDDOC=209966
sameAs	Fano variety Fano variety Fano variety Fano variety Fano variety Fano variety Fano variety Fano variety Fano variety
dbp:wikiPageUsesTemplate	dbt:Authority_control dbt:Citation dbt:Dead_link dbt:Harvtxt dbt:Reflist dbt:Technical dbt:Harvs dbt:Eom
authorlink	Gino Fano (en)
bot	InternetArchiveBot (en)
date	August 2019 (en)
first	Vik.S. (en)
fix-attempted	yes (en)
last	Fano (en) Kulikov (en) Iskovskih (en)
title	Fano_variety (en)
year	1934 (xsd:integer) 1942 (xsd:integer) 1977 (xsd:integer) 1978 (xsd:integer) 1979 (xsd:integer)
has abstract	In der algebraischen Geometrie, einem Teilgebiet der Mathematik, ist eine Fano-Varietät eine vollständige Varietät über einem Körper , deren ist. Eine Fano-Mannigfaltigkeit ist eine singularitäten-freie komplexe Fano-Varietät. (de) In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano , ), is a complete variety X whose anticanonical bundle KX* is ample. In this definition, one could assume that X is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties. (en) 대수기하학에서 파노 다양체(영어: Fano variety)는 사영 공간과 유사하게, 반표준 인자가 풍부한 인자를 이루는 대수다양체이다. (ko)

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