Algebra & Number Theory
- Algebra Number Theory
- Volume 13, Number 2 (2019), 425-454.
Effective generation and twisted weak positivity of direct images
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.
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
Permanent link to this document
Digital Object Identifier
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]
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