Abstract. We prove a strengthening of Kollár's Ampleness Lemma and use it to prove that any proper coarse moduli space of stable log-varieties of general type is projective. We also prove subadditivity of log-Kodaira dimension for fiber spaces whose general fiber is of log general type.

The Chow–Mumford (CM) line bundle is a functorial line bundle on the base of any family of klt Fano varieties. It is conjectured that it yields a polarization on the moduli space of K-poly-stable klt Fano varieties. Proving ampleness of the CM line bundle boils down to showing semi-positivity/positivity statements about the CM-line bundle for families with K-semi-stable/K-polystable fibers. We prove the necessary semi-positivity statements in the K-semi-stable situation, and the necessary positivity statements in the uniform K-stable situation, including in both cases variants assuming K-stability only for general fibers. Our statements work in the most general singular situation (klt singularities), and the proofs are algebraic, except the computation of the limit of a sequence of real numbers via the central limit theorem of probability theory. We also present an application to the classification of Fano varieties. Additionally, our semi-positivity statements work in general for log-Fano pairs.

We establish the Minimal Model Program for arithmetic threefolds whose residue characteristics are greater than five. In doing this, we generalize the theory of global F -regularity to mixed characteristic and identify certain stable sections of adjoint line bundles. Finally, by passing to graded rings, we generalize a special case of Fujita's conjecture to mixed characteristic.

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. CONTENTS[mr] X/Y (mr∆) . This issue will addressed in later articles. Corollary 1.8. In the situation of Notation 1.4, if ∆ = 0, K X/Y is f -ample and for every y ∈ Y , X y is sharply F -pure, Aut(X y ) is finite and there are only finitely many other y ′ ∈ Y such that X y ∼ = X y ′ , then det f * ω [m] X/Y is an ample line bundle for all m ≫ 0 and divisible enough.