Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911

   
 
 

 

Notions of numerical Iitaka dimension do not coincide


Author: John Lesieutre
Journal: J. Algebraic Geom. 31 (2022), 113-126
DOI: https://doi.org/10.1090/jag/763
Published electronically: February 2, 2021
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Abstract: Let $X$ be a smooth projective variety. The Iitaka dimension of a divisor $D$ is an important invariant, but it does not only depend on the numerical class of $D$. However, there are several definitions of “numerical Iitaka dimension”, depending only on the numerical class. In this note, we show that there exists a pseuodoeffective $\mathbb {R}$-divisor for which these invariants take different values. The key is the construction of an example of a pseudoeffective $\mathbb {R}$-divisor $D_+$ for which $h^0(X,\left \lfloor {m D_+}\right \rfloor +A)$ is bounded above and below by multiples of $m^{3/2}$ for any sufficiently ample $A$.


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John Lesieutre
Affiliation: The Pennsylvania State University, 204 McAllister Building, University Park, Pennsylvania 16801
Email: jdl@psu.edu

Received by editor(s): June 5, 2019
Received by editor(s) in revised form: January 16, 2020
Published electronically: February 2, 2021
Additional Notes: This work was supported by NSF Grant DMS-1700898/DMS-1912476.
Article copyright: © Copyright 2021 University Press, Inc.