Fano variety

In algebraic geometry, a Fano variety, introduced in (Fano 1934, 1942), is a complete variety X whose anticanonical bundle K_X^* 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 coherent sheaf cohomology groups Hⁱ(X, O_X) of the structure sheaf vanish for i > 0. It follows that the first Chern class induces an isomorphism c₁: Pic(X) → H²(X, Z).

A smooth complex Fano variety is simply connected. 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. A much easier fact is that every Fano variety has Kodaira dimension −∞.

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