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The dimension of automorphism groups of algebraic varieties with pseudo-effective log canonical divisors


Author: Fei Hu
Journal: Proc. Amer. Math. Soc. 146 (2018), 1879-1893
MSC (2010): Primary 14J50, 14L10, 14L30
DOI: https://doi.org/10.1090/proc/13893
Published electronically: December 4, 2017
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Abstract: Let $ (X,D)$ be a log smooth pair of dimension $ n$, where $ D$ is a reduced effective divisor such that the log canonical divisor $ K_X + D$ is pseudo-effective. Let $ G$ be a connected algebraic subgroup of $ \rm {Aut}(X, D)$. We show that $ G$ is a semi-abelian variety of dimension $ \le \min \{n-\bar {\kappa }(V), n\}$ with $ V\coloneqq X\setminus D$. In the dimension two, Iitaka claimed in his 1979 Osaka J. Math. paper that $ \dim G\le \bar {q}(V)$ for a log smooth surface pair with $ \bar {\kappa }(V) = 0$ and $ \bar {p}_g(V) = 1$. We (re-)prove and generalize this classical result for all surfaces with $ \bar {\kappa }=0$ without assuming Iitaka's classification of logarithmic Iitaka surfaces or logarithmic $ K3$ surfaces.


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Additional Information

Fei Hu
Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076
Address at time of publication: Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC, Canada V6T 1Z2
Email: hf@u.nus.edu

DOI: https://doi.org/10.1090/proc/13893
Keywords: Automorphism, semi-abelian variety, group action, logarithmic Kodaira dimension
Received by editor(s): February 16, 2017
Received by editor(s) in revised form: June 28, 2017, and June 30, 2017
Published electronically: December 4, 2017
Communicated by: Lev Borisov
Article copyright: © Copyright 2017 American Mathematical Society

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