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Effective generation
and twisted weak positivity of direct images
Yajnaseni Dutta and Takumi Murayama |
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Vol. 13 (2019), No. 2, 425–454
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Abstract |
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We study pushforwards of log pluricanonical bundles on projective log canonical pairs 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 surjects onto a projective variety, we show a quadratic bound for generic generation for twists by big and nef line bundles. Second, when 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. |
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Keywords
pluricanonical bundles, Fujita's conjecture, weak
positivity, effective results, Seshadri constants
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Mathematical Subject Classification 2010
Primary: 14C20
Secondary: 14D06, 14F05, 14E30, 14Q20, 14J17
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Milestones
Received: 6 February 2018
Revised: 23 October 2018
Accepted: 24 November 2018
Published: 2 March 2019
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Authors |
| Yajnaseni Dutta | |
| Department of Mathematics Northwestern University Evanston, IL United States |
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| Takumi Murayama | |
| Department of Mathematics University of Michigan Ann Arbor, MI United States |
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