Fano variety

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In algebraic geometry, a Fano variety, introduced in (Fano 1934, 1942), is a 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 ${\mathrm {deg}}(D)<n+1$ .

Some properties

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 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 the del Pezzo surfaces and in the smooth case, they 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 smooth Fano 3-folds with second Betti number 1 into 17 classes, and Mori & Mukai (1981) classified the smooth 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 nulli", 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
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
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

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