#### Vol. 13, No. 2, 2019

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editors' Interests Submission Guidelines Submission Form Editorial Login Ethics Statement Author Index To Appear ISSN: 1944-7833 (e-only) ISSN: 1937-0652 (print) Other MSP Journals
Effective generation and twisted weak positivity of direct images

### Yajnaseni Dutta and Takumi Murayama

Vol. 13 (2019), No. 2, 425–454
DOI: 10.2140/ant.2019.13.425
##### Abstract

We study pushforwards of log pluricanonical bundles on projective log canonical pairs $\left(Y,\Delta \right)$ 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 nonvanishing 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.

##### Keywords
pluricanonical bundles, Fujita's conjecture, weak positivity, effective results, Seshadri constants
##### Mathematical Subject Classification 2010
Primary: 14C20
Secondary: 14D06, 14F05, 14E30, 14Q20, 14J17