On the effective generation of direct images of pluricanonical bundles in mixed characteristic

We present an effective global generation result for direct images of pluricanonical bundles in mixed characteristic. This is a mixed characteristic analog of Ejiri's theorem in positive characteristic and the theorem of Popa and Schnell regarding their Fujita-type conjecture in characteristic zero. As an application, we establish a weak positivity statement for the relative canonical sheaf of a smooth morphism in mixed characteristic. Using this weak positivity result, we prove that images of Fano schemes under smooth morphisms are again Fano in mixed characteristic.