Algebra & Number Theory

Effective generation and twisted weak positivity of direct images

Yajnaseni Dutta and Takumi Murayama

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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.

Article information

Source
Algebra Number Theory, Volume 13, Number 2 (2019), 425-454.

Dates
Received: 6 February 2018
Revised: 23 October 2018
Accepted: 24 November 2018
First available in Project Euclid: 26 March 2019

Permanent link to this document
https://projecteuclid.org/euclid.ant/1553565647

Digital Object Identifier
doi:10.2140/ant.2019.13.425

Mathematical Reviews number (MathSciNet)
MR3927051

Zentralblatt MATH identifier
07042064

Subjects
Primary: 14C20: Divisors, linear systems, invertible sheaves
Secondary: 14D06: Fibrations, degenerations 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20] 14E30: Minimal model program (Mori theory, extremal rays) 14Q20: Effectivity, complexity 14J17: Singularities [See also 14B05, 14E15]

Keywords
pluricanonical bundles Fujita's conjecture weak positivity effective results Seshadri constants

Citation

Dutta, Yajnaseni; Murayama, Takumi. Effective generation and twisted weak positivity of direct images. Algebra Number Theory 13 (2019), no. 2, 425--454. doi:10.2140/ant.2019.13.425. https://projecteuclid.org/euclid.ant/1553565647


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