Vol. 13, No. 2, 2019

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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