Effective generation and twisted weak positivity of direct images

TL;DR

This paper establishes effective bounds for the global generation and weak positivity of pushforwards of log pluricanonical bundles on log canonical pairs, advancing understanding of their positivity properties and providing new vanishing theorems.

Contribution

It introduces new effective global generation and weak positivity results for pushforwards of log pluricanonical bundles in the log canonical setting, partially addressing a Fujita-type conjecture.

Findings

Quadratic bounds for generic generation when $Y$ surjects onto a projective variety.

Linear bounds for twists when $Y$ is fibered over a smooth projective variety.

Effective non-vanishing and vanishing theorems for certain log-sheaves.

Abstract

In this paper, we study pushforwards of log pluricanonical bundles on projective log canonical pairs $(Y,\Delta)$ over the complex numbers, partially answering a Fujita-type conjecture due to Popa and Schnell in the log canonical setting. We show two effective global generation results. First, when $Y$ surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when $Y$ is fibered over a smooth projective variety, we show a linear bound for twists by ample line bundles. These results additionally give effective non-vanishing statements. We also prove an effective weak positivity statement for log pluricanonical bundles in this setting, which may be of independent interest. In each context we indicate over which loci positivity holds. Finally, using the description of such loci, we show an effective vanishing theorem for pushforwards of certain log-sheaves under smooth morphisms.