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| In [[algebraic geometry]], a '''Fano variety''', introduced in {{harvs|authorlink=Gino Fano|last=Fano|year1=1934|year2=1942}}, is a [[Complete algebraic variety|complete variety]] whose [[anticanonical bundle]] is [[ample line bundle|ample]].
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| Fano varieties are quite rare, compared to other families, like [[Calabi–Yau manifold]]s and [[general type surface]]s.
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| ==The example of projective hypersurfaces==
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| The fundamental example of Fano varieties are the [[algebraic geometry of projective spaces|projective spaces]]: the [[canonical bundle|anticanonical line bundle]] of <math>\mathbb P_{\mathbf k}^n</math> is <math>\mathcal O(n+1)</math>, which is [[very ample]] (its [[curvature]] is ''n+1'' times the [[Fubini–Study]] symplectic form).
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| Let ''D'' be a smooth [[Weil divisor]] in <math>\mathbb P_{\mathbf k}^n</math>, from the [[adjunction formula]], we infer <math>\mathcal K_D = (\mathcal K_X + D) = (-(n+1) H + \mathrm{deg}(D) H)_D</math>, where ''H'' is the class of the hyperplane. The [[hypersurface]] ''D'' is therefore Fano if and only if <math>\mathrm{deg}(D) < n+1</math>.
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| ==Some properties==
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| The existence of some ample line bundle on ''X'' is equivalent to ''X'' being a [[projective variety]], so a Fano variety is always projective. The [[Kodaira vanishing theorem]] implies that the [[sheaf cohomology|higher cohomology groups]] <math>H^i(X, \mathcal O_X)</math> of the [[structure sheaf]] vanish for <math>i > 0</math>. In particular, the [[first Chern class]] induces an isomorphism <math>c_1 : \mathrm{Pic}(X) \to H^2(X,\mathbb Z).</math>
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| A Fano variety is simply connected and is [[uniruled]], in particular it has [[Kodaira dimension]] −∞.
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| ==Classification in small dimensions==
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| Fano varieties in dimensions 1 are [[isomorphism|isomorphic]] to the [[projective line]].
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| In dimension 2 they are the [[del Pezzo surface]]s and in the smooth case, they are isomorphic to either <math>\mathbb{P}^1 \times \mathbb{P}^1</math> or to the projective plane blown up in at most 8 general points, and in particular are again all rational.
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| In dimension 3 there are non-rational examples. Iskovskih {{harvtxt|last1=Iskovskih|year1=1977|year2=1978|year3=1979}} classified the smooth Fano 3-folds with second [[Betti number]] 1 into 17 classes, and {{harvtxt|Mori|Mukai|1981}} classified the smooth ones with second Betti number at least 2, finding 88 deformation classes.
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| ==References==
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| *{{citation|first=G. |last=Fano|chapter=Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulli|title= Proc. Internat. Congress Mathematicians (Bologna) , 4 , Zanichelli |year=1934|pages= 115–119}}
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| *{{Citation | doi=10.1007/BF02565618 | last1=Fano | first1=Gino | title=Su alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche | url=http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=209966 | mr=0006445 | year=1942 | journal=Commentarii Mathematici Helvetici | issn=0010-2571 | volume=14 | pages=202–211}}
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| *{{Citation | last1=Iskovskih | first1=V. A. | title=Fano threefolds. I | doi=10.1070/IM1977v011n03ABEH001733 | mr=463151 | year=1977 | journal=Math. USSR-Izv. | issn=0373-2436 | volume=11 | issue=3 | pages=485–527}}
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| *{{Citation | last1=Iskovskih | first1=V. A. | title=Fano 3-folds II | doi=10.1070/IM1978v012n03ABEH001994Fano+3-folds+II | mr=0463151 | year=1978 | journal=Math Ussr Izv | volume=12 | issue=3 | pages=469–506}}
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| *{{Citation | last1=Iskovskih | first1=V. A. | title=Current problems in mathematics, Vol. 12 (Russian) | publisher=VINITI, Moscow | mr=537685 | year=1979 | chapter=Anticanonical models of three-dimensional algebraic varieties | pages=59–157}}
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| *{{eom|id=F/f038220|first=Vik.S.|last= Kulikov}}
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| *{{Citation | last1=Mori | first1=Shigefumi | last2=Mukai | first2=Shigeru | authorlink=Shigefumi Mori | author2-link=Shigeru Mukai | title=Classification of Fano 3-folds with B<sub>2</sub>≥2 | doi=10.1007/BF01170131 | mr=641971 | year=1981 | journal=Manuscripta Mathematica | issn=0025-2611 | volume=36 | issue=2 | pages=147–162}}
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| *{{Citation | last1=Mori | first1=Shigefumi | last2=Mukai | first2=Shigeru | authorlink=Shigefumi Mori | author2-link=Shigeru Mukai | title=Erratum: "Classification of Fano 3-folds with B<sub>2</sub>≥2" | doi=10.1007/s00229-002-0336-2 | mr=1969009 | year=2003 | journal=Manuscripta Mathematica | issn=0025-2611 | volume=110 | issue=3 | pages=407}}
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| [[Category:Algebraic geometry]]
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| [[Category:Threefolds]]
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