Direct images of pluricanonical bundles and Frobenius stable canonical rings of fibers

Abstract

In this paper, we study an algebraic fiber space in positive characteristic whose generic fiber $F$ has finitely generated canonical ring and sufficiently large Frobenius stable canonical ring. An example of such a case is when $F$ is $F$-pure and its dualizing sheaf is invertible and ample. We treat a Fujita-type conjecture due to Popa and Schnell concerning direct images of pluricanonical bundles, and prove it under some additional hypotheses. As an application, we show the subadditivity of Kodaira dimensions in some new cases.

42 pages, v2: Subsection 6.3 added, typos corrected, minor revisions, v3:Definition 5.1 changed, Lemma 5.16 corrected, minor revisions, v4:Introduction revised, Definitions 3.3 and 3.4 changed, Lemma 3.7, Theorems 6.7, 6.9 and 6.11 corrected, v5:References and typos corrected

References in corpus (1)

• On mixed-$\omega$-sheaves

Cited by in corpus (1)

• Subadditivity of Kodaira dimension does not hold in positive characteristic