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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

$ n$-dimensional projective varieties with the action of an abelian group of rank $ n-1$


Author: De-Qi Zhang
Journal: Trans. Amer. Math. Soc. 368 (2016), 8849-8872
MSC (2010): Primary 32H50, 14J50, 32M05, 11G10, 37B40
DOI: https://doi.org/10.1090/tran/6629
Published electronically: February 12, 2016
MathSciNet review: 3551591
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Abstract: Let $ X$ be a normal projective variety of dimension $ n \ge 3$ admitting the action of the group $ G := \mathbb{Z}^{\oplus n-1}$ such that every non-trivial element of $ G$ is of positive entropy. We show: `$ X$ is not rationally connected' $ \Rightarrow $ `$ X$ is $ G$-equivariant birational to the quotient of a complex torus' $ \Leftarrow \Rightarrow $ `$ K_X + D$ is pseudo-effective for some $ G$-periodic effective fractional divisor $ D$'. To apply, one uses the above and the fact: `the Kodaira dimension $ \kappa (X) \ge 0$' $ \Rightarrow $ `$ X$ is not uniruled' $ \Rightarrow $ `$ X$ is not rationally connected'. We may generalize the result to the case of solvable $ G$.


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Additional Information

De-Qi Zhang
Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076
Email: matzdq@nus.edu.sg

DOI: https://doi.org/10.1090/tran/6629
Keywords: Automorphism, iteration, complex dynamics, tori, topological entropy
Received by editor(s): July 1, 2014
Received by editor(s) in revised form: November 21, 2014
Published electronically: February 12, 2016
Additional Notes: The author was supported by an ARF of NUS
Article copyright: © Copyright 2016 American Mathematical Society

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