Algebra & Number Theory

Effective generation and twisted weak positivity of direct images

Yajnaseni Dutta and Takumi Murayama

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]

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


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.

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