ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Sun, 14 Jul 2024 10:41:11 GMT2024-07-14T10:41:11Z5031Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectureshttps://scholarbank.nus.edu.sg/handle/10635/234643Title: Endomorphisms of quasi-projective varieties -- towards Zariski dense orbit and Kawaguchi-Silverman conjectures
Authors: Jia, Jia; Shibata, Takahiro; Xie, Junyi; Zhang, De-Qi
Abstract: Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite
surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and
Adelic Zariski Dense Orbit Conjecture (AZO). We consider also
Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical
degree $d_1(f)$ of $f$ equals the arithmetic degree $\alpha_f(P)$ at a point
$P$ having Zariski dense $f$-forward orbit. Assuming $X$ is a smooth affine
surface, such that the log Kodaira dimension $\bar{\kappa}(X)$ is non-negative
(resp. the \'etale fundamental group $\pi_1^{\text{\'et}}(X)$ is infinite), we
confirm AZO, (hence) ZDO, and KSC (when $\operatorname{deg}(f)\geq 2$) (resp.
AZO and hence ZDO). We also prove ZDO (resp. AZO and hence ZDO) for every
surjective endomorphism on any projective variety with ''larger'' first
dynamical degree (resp. every dominant endomorphism of any semiabelian
variety).
Mon, 12 Apr 2021 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346432021-04-12T00:00:00ZNon-density of points of small arithmetic degreeshttps://scholarbank.nus.edu.sg/handle/10635/234648Title: Non-density of points of small arithmetic degrees
Authors: Matsuzawa, Yohsuke; Meng, Sheng; Shibata, Takahiro; Zhang, De-Qi
Abstract: Given a surjective endomorphism $f: X \to X$ on a projective variety over a
number field, one can define the arithmetic degree $\alpha_f(x)$ of $f$ at a
point $x$ in $X$. The Kawaguchi--Silverman Conjecture (KSC) predicts that any
forward $f$-orbit of a point $x$ in $X$ at which the arithmetic degree
$\alpha_f(x)$ is strictly smaller than the first dynamical degree $\delta_f$ of
$f$ is not Zariski dense. We extend the KSC to sAND (= small Arithmetic
Non-Density) Conjecture that the locus $Z_f(d)$ of all points of small
arithmetic degree is not Zariski dense, and verify this sAND Conjecture for
endomorphisms on projective varieties including surfaces, HyperK\"ahler
varieties, abelian varieties, Mori dream spaces, simply connected smooth
varieties admitting int-amplified endomorphisms, smooth threefolds admitting
int-amplified endomorphisms, and some fiber spaces. We also show close
relations between our sAND Conjecture and the Uniform Boundedness Conjecture of
Morton and Silverman on endomorphisms of projective spaces and another long
standing conjecture on Uniform Boundedness of torsion points in abelian
varieties.
Tue, 25 Feb 2020 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346482020-02-25T00:00:00ZInvariant Subvarieties With Small Dynamical Degreehttps://scholarbank.nus.edu.sg/handle/10635/234618Title: Invariant Subvarieties With Small Dynamical Degree
Authors: Matsuzawa, Y; Meng, S; Shibata, T; Zhang, DQ; Zhong, G
Abstract: Let f : X → X be a dominant self-morphism of an algebraic variety. Consider the set ∑f8 of f -periodic subvarieties of small dynamical degree (SDD), the subset Sf8 of maximal elements in ∑f8, and the subset Sf of f -invariant elements in Sf8. When X is projective, we prove the finiteness of the set Pf of f -invariant prime divisors with SDD and give an optimal upper bound Pf n = d1(f )n(1 + o(1)) as n→8, where d1(f ) is the 1st dynamic degree. When X is an algebraic group (with f being a translation of an isogeny), or a (not necessarily complete) toric variety, we give an optimal upper bound Sf n = d1(f )n dim(X)(1 + o(1)) as n→8, which slightly generalizes a conjecture of S.-W. Zhang for polarized f .
Fri, 01 Jul 2022 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346182022-07-01T00:00:00Z