Fano Manifold

A Fano manifold is a smooth connected projective complex variety whose anticanonical bundle is ample. Projective space is an example, since its anticanonical bundle is . Rational curves on a positive-dimensional Fano manifold define its pseudoindex, the smallest anticanonical degree of an irreducible rational curve.

The Picard number measures the dimension of its numerical divisor-class space. The generalized Mukai conjecture bounds the product of this number with one less than the pseudoindex. The pseudoindex uses curve degrees, whereas the Fano index uses divisibility in the Picard group.