On Polarized Manifolds Whose Adjoint Bundles Are Not Semipositive

Homological Algebra of Mirror Symmetry

“…(2.6) Conjecture (Fujita, 1987). If L is an ample line bundle on a projective n-fold X, then K X + (n + 1)L is globally generated and K X + (n + 2)L is very ample.…”

Characterization of projective spaces by Seshadri constants

“…Using Kodaira vanishing theorem, Kobayashi and Ochiai [14] proved that if an n-dimensional projective manifold X with an ample line bundle H satisfies −K X ≡ (n + 1)H , then (X, H ) ∼ = (P n , O(1)). Kobayashi-Ochiai's characterization was generalized by Ionescu [11] (in the smooth case) and Fujita [8] (allowing Gorenstein rational singularities) assuming the weaker condition that K X + (n + 1)H is not ample. Later, Cho, Miyaoka and ShepherdBarron [5] (simplified by Kebekus in [13]) showed that a Fano manifold is isomorphic to P n if the anti-canonical degree of every curve is at least n + 1.…”

Ruled Fano fivefolds of index two

“…By lemma 3.7, there exists an ample L ∈ Pic(X) such that ψ ϑ is supported by K X + rL, and we conclude by [22,Lemma 2.12].…”

“…By adjunction the general fiber of p is a projective space of dimension dim X − 2; over a general hyperplane section of S, ϕ is a projective bundle by lemma 2.17, whence the locus over which the fiber is not a projective space is discrete in S. We can apply [5,Lemma 3.3] and [22,Lemma 2.12] to obtain that every fiber of ϕ is a projective space. The surface S is dominated by a Fano manifold, hence is rationally connected; therefore H 2 (S, O * ) = 0 and the Brauer group of S is trivial.…”

“…An equidimensional scroll is a projective bundle by [22,Lemma 2.12], while an equidimensional quadric fibration is a quadric bundle by [3,Theorem B].…”