Abstract
We prove that an n-dimensional complex projective variety is isomorphic to {mathbb {P}}^n if the Seshadri constant of the anti-canonical divisor at some smooth point is greater than n. We also classify complex projective varieties with Seshadri constants equal to n.
Highlights
It is believed that the projective space Pn has the most positive anti-canonical divisor among complex projective varieties
Cho, Miyaoka and ShepherdBarron [5] showed that a Fano manifold is isomorphic to Pn if the anti-canonical degree of every curve is at least n + 1
Their proofs rely on deformation of rational curves which still work if we allow isolated local complete intersection quotient singularities
Summary
It is believed that the projective space Pn has the most positive anti-canonical divisor among complex projective varieties. Cho, Miyaoka and ShepherdBarron [5] (simplified by Kebekus in [13]) showed that a Fano manifold is isomorphic to Pn if the anti-canonical degree of every curve is at least n + 1 Their proofs rely on deformation of rational curves which still work if we allow isolated local complete intersection quotient singularities (see [4]). The purpose of this paper is to provide a characterization of Pn among complex Q-Fano varieties by the local positivity of the anti-canonical divisor, namely the Seshadri constants. Theorem 4 For any rational number 0 < c ≤ n, there exists an n-dimensional Q-Fano variety X with a smooth point p such that (−K X , p) = c.
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