A Conjecture of Mukai Relating Numerical Invariants of Fano Manifolds

摘要

A complex manifold X of dimension n such that the anticanonical bundle –K X := det TX is ample is called a Fano manifold. Besides the dimension, other two integers play an essential role in the classification of these manifolds, namely the pseudoindex of X, i X , which is the minimal anticanonical degree of rational curves on X, and the Picard number ρ X , the dimension of N 1(X), the vector space generated by irreducible complex curves modulo numerical equivalence . A (generalization of a) conjecture of Mukai says that ρ X (i X – 1) ≤ n. In this paper we present some partial steps towards the conjecture, we show how one can interpretate and possibly solve it with the use of families of rational curves on a uniruled variety, and more generally with the instruments of Mori theory. We consider also other related problems: the description of some Fano manifolds which are at the border of the Mukai relations and how the pseudoindex changes via (some) birational transformation.