ScholarBank@NUShttps://scholarbank.nus.edu.sgThe DSpace digital repository system captures, stores, indexes, preserves, and distributes digital research material.Mon, 22 Jul 2024 15:40:53 GMT2024-07-22T15:40:53Z50121Characterizations of toric varieties via polarized endomorphismshttps://scholarbank.nus.edu.sg/handle/10635/234653Title: Characterizations of toric varieties via polarized endomorphisms
Authors: Meng, Sheng; Zhang, De-Qi
Abstract: Let X be a normal projective variety and f: X→ X a non-isomorphic polarized endomorphism. We give two characterizations for X to be a toric variety. First we show that if X is Q-factorial and G-almost homogeneous for some linear algebraic group G such that f is G-equivariant, then X is a toric variety. Next we give a geometric characterization: if X is of Fano type and smooth in codimension 2 and if there is an f- 1-invariant reduced divisor D such that f| X\D is quasi-étale and KX+ D is Q-Cartier, then X admits a quasi-étale cover X~ such that X~ is a toric variety and f lifts to X~. In particular, if X is further assumed to be smooth, then X is a toric variety.
Thu, 01 Aug 2019 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346532019-08-01T00:00:00ZJORDAN PROPERTY FOR NON-LINEAR ALGEBRAIC GROUPS AND PROJECTIVE VARIETIEShttps://scholarbank.nus.edu.sg/handle/10635/234655Title: JORDAN PROPERTY FOR NON-LINEAR ALGEBRAIC GROUPS AND PROJECTIVE VARIETIES
Authors: Meng, Sheng; Zhang, De-Qi
Abstract: A century ago, Camille Jordan proved that the complex general linear group GLn(ℂ) has the Jordan property: there is a Jordan constant Cn such that every finite subgroup H ≤ GLn(ℂ) has an abelian subgroup H1 of index [H: H1] ≤ Cn. We show that every connected algebraic group G (which is not necessarily linear) has the Jordan property with the Jordan constant depending only on dim G, and that the full automorphism group Aut(X) of every projective variety X has the Jordan property.
Wed, 01 Aug 2018 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346552018-08-01T00:00:00ZPolarized endomorphisms of normal projective threefolds in arbitrary characteristichttps://scholarbank.nus.edu.sg/handle/10635/234646Title: Polarized endomorphisms of normal projective threefolds in arbitrary characteristic
Authors: Cascini, Paolo; Meng, Sheng; Zhang, De-Qi
Abstract: Let X be a projective variety over an algebraically closed field k of arbitrary characteristic p≥ 0. A surjective endomorphism f of X is q-polarized if f∗H∼ qH for some ample Cartier divisor H and integer q> 1. Suppose f is separable and X is Q-Gorenstein and normal. We show that the anti-canonical divisor - KX is numerically equivalent to an effective Q-Cartier divisor, strengthening slightly the conclusion of Boucksom, de Fernex and Favre (Duke Math J 161(8):1455–1520, 2012, Theorem C) and also covering singular varieties over an algebraically closed field of arbitrary characteristic. Suppose f is separable and X is normal. We show that the Albanese morphism of X is an algebraic fibre space and f induces polarized endomorphisms on the Albanese and also the Picard variety of X, and KX being pseudo-effective and Q-Cartier means being a torsion Q-divisor. Let fGal: X¯ → X be the Galois closure of f. We show that if p> 5 and co-prime to deg fGal then one can run the minimal model program (MMP) f-equivariantly, after replacing f by a positive power, for a mildly singular threefold X and reach a variety Y with torsion canonical divisor (and also with Y being a quasi-étale quotient of an abelian variety when dim (Y) ≤ 2). Along the way, we show that a power of f acts as a scalar multiplication on the Neron-Severi group of X (modulo torsion) when X is a smooth and rationally chain connected projective variety of dimension at most three.
Thu, 01 Oct 2020 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346462020-10-01T00:00:00ZJordan property for automorphism groups of compact spaces in Fujiki's class $\mathcal{C}$https://scholarbank.nus.edu.sg/handle/10635/234645Title: Jordan property for automorphism groups of compact spaces in Fujiki's class $\mathcal{C}$
Authors: Meng, Sheng; Perroni, Fabio; Zhang, De-Qi
Abstract: Let $X$ be a compact complex space in Fujiki's Class $\mathcal{C}$. We show
that the group $Aut(X)$ of all biholomorphic automorphisms of $X$ has the
Jordan property: there is a (Jordan) constant $J = J(X)$ such that any finite
subgroup $G\le Aut(X)$ has an abelian subgroup $H\le G$ with the index
$[G:H]\le J$. This extends, with a quite different method, the result of
Prokhorov and Shramov for Moishezon threefolds.
Thu, 19 Nov 2020 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346452020-11-19T00:00:00ZBuilding blocks of polarized endomorphisms of normal projective varietieshttps://scholarbank.nus.edu.sg/handle/10635/234660Title: Building blocks of polarized endomorphisms of normal projective varieties
Authors: Meng, Sheng; Zhang, De-Qi
Abstract: An endomorphism f of a projective variety X is polarized (resp. quasi-polarized) if f⁎H∼qH (linear equivalence) for some ample (resp. nef and big) Cartier divisor H and integer q>1. First, we use cone analysis to show that a quasi-polarized endomorphism is always polarized, and the polarized property descends via any equivariant dominant rational map. Next, we show that a suitable maximal rationally connected fibration (MRC) can be made f-equivariant using a construction of N. Nakayama, that f descends to a polarized endomorphism of the base Y of this MRC and that this Y is a Q-abelian variety (quasi-étale quotient of an abelian variety). Finally, we show that we can run the minimal model program (MMP) f-equivariantly for mildly singular X and reach either a Q-abelian variety or a Fano variety of Picard number one. As a consequence, the building blocks of polarized endomorphisms are those of Q-abelian varieties and those of Fano varieties of Picard number one. Along the way, we show that f always descends to a polarized endomorphism of the Albanese variety Alb(X) of X, and that the pullback of a power of f acts as a scalar multiplication on the Néron–Severi group of X (modulo torsion) when X is smooth and rationally connected. Partial answers about X being of Calabi–Yau type, or Fano type are also given with an extra primitivity assumption on f which seems necessary by an example.
Mon, 05 Feb 2018 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346602018-02-05T00:00:00ZBuilding blocks of amplified endomorphisms of normal projective varietieshttps://scholarbank.nus.edu.sg/handle/10635/234659Title: Building blocks of amplified endomorphisms of normal projective varieties
Authors: Meng, Sheng
Abstract: Let $X$ be a normal projective variety. A surjective endomorphism $f:X\to X$
is int-amplified if $f^\ast L - L =H$ for some ample Cartier divisors $L$ and
$H$. This is a generalization of the so-called polarized endomorphism which
requires that $f^*H\sim qH$ for some ample Cartier divisor $H$ and $q>1$. We
show that this generalization keeps all nice properties of the polarized case
in terms of the singularity, canonical divisor, and equivariant minimal model
program.
Mon, 25 Dec 2017 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346592017-12-25T00:00:00ZSemi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphismhttps://scholarbank.nus.edu.sg/handle/10635/234651Title: Semi-group structure of all endomorphisms of a projective variety admitting a polarized endomorphism
Authors: Meng, Sheng; Zhang, De-Qi
Abstract: Let X be a projective variety admitting a polarized (or more generally, int-amplified) endomorphism. We show: there are only finitely many contractible extremal rays; and when X is Q-factorial normal, every minimal model program is equivariant relative to the monoid SEnd(X) of all surjective endomorphisms, up to finite index. Further, when X is rationally connected and smooth, we show: there is a finite-index submonoid G of SEnd(X) such that G acts via pullback as diagonal (and hence commutative) matrices on the Neron-Severi group; the full automorphisms group Aut(X) has finitely many connected components; and every amplified endomorphism is int-amplified.
Wed, 01 Jan 2020 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346512020-01-01T00:00:00ZKawaguchi-Silverman conjecture for surjective endomorphismshttps://scholarbank.nus.edu.sg/handle/10635/234652Title: Kawaguchi-Silverman conjecture for surjective endomorphisms
Authors: Meng, Sheng; Zhang, De-Qi
Abstract: We prove the Kawaguchi-Silverman conjecture (KSC), about the equality of
arithmetic degree and dynamical degree, for every surjective endomorphism of
any (possibly singular) projective surface. In high dimensions, we show that
KSC holds for every surjective endomorphism of any $\mathbb{Q}$-factorial
Kawamata log terminal projective variety admitting an int-amplified
endomorphism, provided that KSC holds for any surjective endomorphism with the
ramification divisor being totally invariant and irreducible. In particular, we
show that KSC holds for every surjective endomorphism of any rationally
connected smooth projective threefold admitting an int-amplified endomorphism.
The main ingredients are the equivariant minimal model program, the
effectiveness of the anti-canonical divisor and a characterization of toric
pairs.
Mon, 05 Aug 2019 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346522019-08-05T00:00:00ZNon-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/10635/2346182022-07-01T00:00:00ZNon-isomorphic endomorphisms of Fano threefoldshttps://scholarbank.nus.edu.sg/handle/10635/234619Title: Non-isomorphic endomorphisms of Fano threefolds
Authors: Meng, Sheng; Zhang, De-Qi; Zhong, Guolei
Abstract: Let X be a smooth Fano threefold. We show that X admits a non-isomorphic surjective endomorphism if and only if X is either a toric variety or a product of P1 and a del Pezzo surface; in this case, X is a rational variety. We further show that X admits a polarized (or amplified) endomorphism if and only if X is a toric variety.
Thu, 30 Sep 2021 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346192021-09-30T00:00:00ZNormal Projective Varieties Admitting Polarized or Int-amplified Endomorphismshttps://scholarbank.nus.edu.sg/handle/10635/234649Title: Normal Projective Varieties Admitting Polarized or Int-amplified Endomorphisms
Authors: Meng, S; Zhang, DQ
Abstract: Let X be a normal projective variety admitting a polarized or int-amplified endomorphism f. We list up characteristic properties of such an endomorphism and classify such a variety from the aspects of its singularity, anti-canonical divisor, and Kodaira dimension. Then, we run the equivariant minimal model program with respect to not just the single f but also the monoid SEnd(X) of all surjective endomorphisms of X, up to finite-index. Several applications are given. We also give both algebraic and geometric characterizations of toric varieties via polarized endomorphisms.
Sun, 01 Mar 2020 00:00:00 GMThttps://scholarbank.nus.edu.sg/handle/10635/2346492020-03-01T00:00:00Z