Fano variety

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

The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of $\mathbb P_{\mathbf k}^n$ is $\mathcal O(n+1)$, which is very ample (its curvature is n+1 times the Fubini–Study symplectic form).

Let D be a smooth Weil divisor in $\mathbb P_{\mathbf k}^n$, from the adjunction formula, we infer $\mathcal K_D = (\mathcal K_X + D) = -(n+1) H + \mathrm{deg}(D) H )_D$, where H is the class of the hyperplane. The hypersurface D is therefore Fano if and only if D < n + 1.

Some properties

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 $H^i(X, \mathcal O_X)$ of the structure sheaf vanish for i > 0. In particular, the first Chern class induces an isomorphism $c_1 : \mathrm{Pic}(X) \to H^2(X,\mathbb Z).$

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

Classification in small dimensions

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

In dimension 2 they are del Pezzo surfaces and are isomorphic to either $\mathbb{P}^1 \times \mathbb{P}^1$ 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

• 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, MR0006445, http://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, MR463151 
• Iskovskih, V. A. (1978), "Fano 3-folds II", Math Ussr Izv 12 (3): 469–506, doi:10.1070/IM1978v012n03ABEH001994Fano+3-folds+II, MR0463151 
• Iskovskih, V. A. (1979), "Anticanonical models of three-dimensional algebraic varieties", Current problems in mathematics, Vol. 12 (Russian), VINITI, Moscow, pp. 59–157, MR537685 
• Kulikov, Vik.S. (2001), "Fano variety", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1556080104, http://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, MR641971 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, MR1969009