Fano variety

In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano 1934, 1942), 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. In algebraic geometry, a Fano variety, introduced by Gino Fano in (Fano 1934, 1942), 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. The existence of some ample line bundle on X is equivalent to X being a projective variety, so a Fano variety is always projective. For a Fano variety X over the complex numbers, the Kodaira vanishing theorem implies that the sheaf cohomology groups H j ( X , O X ) {displaystyle H^{j}(X,{mathcal {O}}_{X})} of the structure sheaf vanish for j > 0 {displaystyle j>0} . In particular, the Todd genus χ ( X , O ) = ∑ ( − 1 ) j h j ( X , O X ) {displaystyle chi (X,{mathcal {O}})=sum (-1)^{j}h^{j}(X,{mathcal {O}}_{X})} automatically equals 1. The j = 1 , 2 {displaystyle j=1,2} cases of this vanishing statement also tell us that the first Chern class induces an isomorphism c 1 : P i c ( X ) → H 2 ( X , Z ) {displaystyle c_{1}:Pic(X) o H^{2}(X,mathbb {Z} )} . By Yau's solution of the Calabi conjecture, a smooth complex variety admits Kähler metrics of positiveRicci curvature if and only if it is Fano. Myers' theorem therefore tells us that the universal cover of a Fano manifold is compact, and so can only be a finite covering. However, we have just seen that the Todd genus of a Fano manifold must equal 1. Since this would also apply to the manifold's universal cover, and since the Todd genus is multiplicative under finite covers, it follows that any Fano manifold is simply connected. A much easier fact is that every Fano variety has Kodaira dimension −∞. Campana and Kollár–Miyaoka–Mori showed that a smooth Fano variety over an algebraically closed field is rationally chain connected; that is, any two closed points can be connected by a chain of rational curves.Kollár–Miyaoka–Mori also showed that the smooth Fano varieties of a given dimension over an algebraically closed field of characteristic zero form a bounded family, meaning that they are classified by the points of finitely many algebraic varieties. In particular, there are only finitely many deformation classes of Fano varieties of each dimension. In this sense, Fano varieties are much more special than other classes of varieties such as varieties of general type.