Effective generation and twisted weak positivity of direct images

Dutta, Yajnaseni; Murayama, Takumi

doi:10.2140/ant.2019.13.425

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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 $(Y, Δ)$ 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

Milestones

Received: 6 February 2018

Revised: 23 October 2018

Accepted: 24 November 2018

Published: 2 March 2019

Authors

Yajnaseni Dutta
Department of Mathematics Northwestern University Evanston, IL United States

Takumi Murayama
Department of Mathematics University of Michigan Ann Arbor, MI United States

Vol. 13, No. 2, 2019