$n$-dimensional projective varieties with the action of an abelian group of rank $n-1$
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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 $G$.References
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Additional Information
- De-Qi Zhang
- Affiliation: Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore 119076
- MR Author ID: 187025
- ORCID: 0000-0003-0139-645X
- Email: matzdq@nus.edu.sg
- 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
- © Copyright 2016 American Mathematical Society
- 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
- MathSciNet review: 3551591