Semi-positivity in positive characteristics

Abstract

Let f : (X, ∆) → Y be a flat, projective family of sharply F -pure, log-canonically polarized pairs over an algebraically closed field of characteristic p > 0 such that p ∤ ind(K X/Y + ∆).We show that K X/Y + ∆ is nef and that f * (O X (m(K X/Y + ∆))) is a nef vector bundle for m ≫ 0 and divisible enough.Some of the results also extend to non log-canonically polarized pairs.The main motivation of the above results is projectivity of proper subspaces of the moduli space of stable pairs in positive characteristics.Other applications are Kodaira vanishing free, algebraic proofs of corresponding positivity results in characteristic zero, and special cases of subadditivity of Kodairadimension in positive characteristics.REMARK 2.6.It is very important to stress that S 0 (X, σ(X, ∆) ⊗ L ) depends on ∆ and L , not only on σ(X, ∆) ⊗ L and not even on σ(X, ∆) and L .The following proposition gives a better description of S 0 (X, σ(X, ∆) ⊗ L ).It is the one that will be used throughout the article.Proposition 2.7.In the situation of Notation 2.2, if L is a line bundle on X thenH 0 (X, L )).Proof.The difference between the left side of (2.5.a) and (2.7.a) is that the σ(X, F ) is omitted from the latter one.Hence the latter is bigger and in particular using (2.5.a),To prove the other inclusion, consider then the following isomorphisms for any g|e.F e * (σ(X, F )⊗L e,∆ ) ∼ = F e * (φ e ′ F e ′ * L e ′ ,∆ ) ⊗ L e,∆ by (2.4.a), for e ′ ≫ 0 such that g|e ′ ∼ = F e * φ e ′ ⊗ id L e,∆ F e ′ * L e ′ ,∆ ⊗ L p e ′ e,∆ projection formula ∼ = F e * (φ e ′ ⊗ id L e,∆ (F e ′ * L e+e ′ ,∆ )) ∼ = F e * (φ e ′ ⊗ id L e,∆ )(F e+e ′ * L e+e ′ ,∆ )This yields a homomorphism H 0 (X, F e+e ′

Locations

• Annales Scientifiques de l École Normale Supérieure - View
• arXiv (Cornell University) - View - PDF