Semi-positivity in positive characteristics

Abstract

Let $f:(X,\Delta )\to Y$ be a flat, projective family of sharply $F$-pure, log-canonically polarized pairs over an algebraically closed field of characteristic $p>0$ such that $p \nmid \ind(K_{X/Y} + Δ)$. We show that $K_{X/Y}+\Delta$ is nef and that $f_* (\sO_X(m (K_{X/Y} + Δ)))$ is a nef vector bundle for $m\gg 0$ 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 Kodaira-dimension in positive characteristics.

34 pages, comments are welcomed

References in corpus (3)

• Supersingular K3 surfaces for large primes
• Moduli of varieties of general type
• Depth of $F$-singularities and base change of relative canonical sheaves

Cited by in corpus (5)

• Test ideals of non-principal ideals: Computations, Jumping Numbers, Alterations and Division Theorems
• On the three dimensional minimal model program in positive characteristic
• F-singularities in families
• Projectivity of the moduli space of stable log-varieties and subadditvity of log-Kodaira dimension
• The subadditivity of the Kodaira Dimension for Fibrations of Relative Dimension One in Positive Characteristics