On the global generation of higher direct images of pluricanonical bundles

TL;DR

This paper investigates the conditions under which higher direct images of pluricanonical bundles are globally generated in the context of fibrations between projective manifolds, addressing open questions in the field.

Contribution

It provides new results on the effective generation of higher direct images of pluricanonical bundles, answering specific questions posed by Popa and Schnell.

Findings

Established effective criteria for global generation of higher direct images

Answered two open questions by Popa and Schnell

Enhanced understanding of the behavior of pluricanonical bundles under fibrations

Abstract

Given a fibration $f$ between two projective manifolds $X$ and $Y$, we discuss the effective generation of the higher direct images $R^{i}f_{\ast}(K^{m}_{X})$, where $K^{m}_{X}$ is the $m$-th tensor power of the canonical bundle of $X$. In particular, we answer two questions posed by Popa--Schnell in [PS14].