In algebraic geometry, a Fano variety, introduced by (Fano 1934, 1942), is a non-singular complete variety whose anticanonical bundle is ample.

Fano varieties are quite rare, compared to other families, like Calabi–Yau manifolds and general type surfaces.

The example of projective hypersurfaces [link]

The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb P_{\mathbf k}^n

is Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal O(n+1)

, which is very ample (its curvature is n+1 times the Fubini–Study symplectic form).

Let D be a smooth Weil divisor in Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb P_{\mathbf k}^n , from the adjunction formula, we infer Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal K_D = (\mathcal K_X + D) = (-(n+1) H + \mathrm{deg}(D) H)_D , where H is the class of the hyperplane. The hypersurface D is therefore Fano if and only if Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{deg}(D) < n+1 .

Some properties [link]

The existence of an ample line bundle on X is equivalent to X being a projective variety, so this is the case for Fano varieties. The Kodaira vanishing theorem implies that the higher cohomology groups Failed to parse (Missing texvc executable; please see math/README to configure.): H^i(X, \mathcal O_X)

of the structure sheaf vanish for Failed to parse (Missing texvc executable; please see math/README to configure.): i > 0

. In particular, the first Chern class induces an isomorphism Failed to parse (Missing texvc executable; please see math/README to configure.): c_1 : \mathrm{Pic}(X) \to H^2(X,\mathbb Z).

A Fano variety is simply connected and is uniruled, in particular it has Kodaira dimension −∞.

Classification in small dimensions [link]

Fano varieties in dimensions 1 are isomorphic to the projective line.

In dimension 2 they are del Pezzo surfaces and are isomorphic to either Failed to parse (Missing texvc executable; please see math/README to configure.): \mathbb{P}^1 \times \mathbb{P}^1

or to the projective plane blown up in at most 8 general points, and in particular are again all rational. 

In dimension 3 there are non-rational examples. Iskovskih () classified the Fano 3-folds with second Betti number 1 into 17 classes, and Mori & Mukai (1981) classified the ones with second Betti number at least 2, finding 88 deformation classes.

References [link]

• Fano, G. (1934), "Sulle varietà algebriche a tre dimensioni aventi tutti i generi nulu", Proc. Internat. Congress Mathematicians (Bologna) , 4 , Zanichelli, pp. 115–119 
• Fano, Gino (1942), "Su alcune varietà algebriche a tre dimensioni razionali, e aventi curve-sezioni canoniche", Commentarii Mathematici Helvetici 14: 202–211, DOI:10.1007/BF02565618, ISSN 0010-2571, MR 0006445, https://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=209966 
• Iskovskih, V. A. (1977), "Fano threefolds. I", Math. USSR-Izv. 11 (3): 485–527, DOI:10.1070/IM1977v011n03ABEH001733, ISSN 0373-2436, MR 463151 
• Iskovskih, V. A. (1978), "Fano 3-folds II", Math Ussr Izv 12 (3): 469–506, DOI:10.1070/IM1978v012n03ABEH001994Fano+3-folds+II, MR 0463151 
• Iskovskih, V. A. (1979), "Anticanonical models of three-dimensional algebraic varieties", Current problems in mathematics, Vol. 12 (Russian), VINITI, Moscow, pp. 59–157, MR 537685 
• Kulikov, Vik.S. (2001), "F/f038220", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=F/f038220 
• Mori, Shigefumi; Mukai, Shigeru (1981), "Classification of Fano 3-folds with B₂≥2", Manuscripta Mathematica 36 (2): 147–162, DOI:10.1007/BF01170131, ISSN 0025-2611, MR 641971 Mori, Shigefumi; Mukai, Shigeru (2003), "Erratum: "Classification of Fano 3-folds with B₂≥2"", Manuscripta Mathematica 110 (3): 407, DOI:10.1007/s00229-002-0336-2, ISSN 0025-2611, MR 1969009