相关论文: Building blocks of polarized endomorphisms of norm…

Let $X$ be a projective variety over an algebraically closed field $k$ of arbitrary characteristic $p \ge 0$. A surjective endomorphism $f$ of $X$ is $q$-polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q >…

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…

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 $\mathbb{Q}$-factorial normal, every minimal model…

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. It was conjectured by Broustet and Gongyo that $X$ is of Calabi-Yau type, i.e., $(X,\Delta)$ is…

We consider an arbitrary int-amplified surjective endomorphism $f$ of a normal projective variety $X$ over $\mathbb{C}$ and its $f^{-1}$-stable prime divisors. We extend the early result for the case of polarized endomorphisms to the case…

We show that polarized endomorphisms of rationally connected threefolds with at worst terminal singularities are equivariantly built up from those on Q-Fano threefolds, Gorenstein log del Pezzo surfaces and P^1. Similar results are obtained…

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…

Let $X$ be a klt projective variety with numerically trivial canonical divisor. A surjective endomorphism $f:X\to X$ is amplified (resp.~quasi-amplified) if $f^*D-D$ is ample (resp.~big) for some Cartier divisor $D$. We show that after…

Let $X$ be a normal projective variety admitting a polarized endomorphism $f$, i.e., $f^*H\sim qH$ for some ample divisor $H$ and integer $q>1$. Then Broustet and Gongyo proposed the conjecture that $X$ is of Calabi-Yau type (CY for short),…

It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described by methods of equivariant lifting and…

Let $X$ be a normal projective variety and $f:X\to X$ a non-isomorphic polarized endomorphism. We give two characterizations for $X$ to be a toric variety. First we show that if $X$ is $\mathbb{Q}$-factorial and $G$-almost homogeneous for…

Let $X$ be a Fano type variety and $(X,\Delta)$ be a log Calabi-Yau pair with $\Delta$ a Weil divisor. If $(X,\Delta)$ admits a polarized endomorphism, then we show that $(X,\Delta)$ is a finite quotient of a toric pair. Along the way, we…

Let X be a normal variety such that $K_X$ is Q-Cartier, and let $f: X \rightarrow X$ be a finite surjective morphism of degree at least two. We establish a close relation between the irreducible components of the locus of singularities that…

We study finite quotients of abelian varieties (fqav for short) i.e. quotients of abelian varieties by finite groups. We show that Q-abelian varieties (i.e. fqav's with $\mathbb Q$-linearly trivial canonical divisors) are characterized by…

In this paper, we study the structure of Fano fibrations of varieties admitting an int-amplified endomorphism. We prove that if a normal $\mathbb{Q}$-factorial klt projective variety $X$ has an int-amplified endomorphism, then there exists…

We show that assuming the standard conjectures, for any smooth projective variety $X$ of dimension $n$ over an algebraically closed field, there is a constant $C>0$ such that for any positive rational number $r$ and for any polarized…

Let $f:X\to X$ be a non-isomorphic (i.e., $\text{deg } f>1$) surjective endomorphism of a smooth projective threefold $X$. We prove that any birational minimal model program becomes $f$-equivariant after iteration, provided that $f$ is…

We consider surjective endomorphisms f of degree > 1 on projective manifolds X of Picard number one and their f^{-1}-stable hypersurfaces V, and show that V is rationally chain connected. Also given is an optimal upper bound for the number…

Let $X$ be a smooth projective variety defined on a finite field $\mathbb{F}_q$. On $X$ there is a special morphism $Fr_X$, which raises coordinates to exponent $q$: $t\mapsto t^q$. The two main results in this paper are: Result 1: If…

Let (X,D) be a dlt pair, where X is a normal projective variety. Let S denote the support of the rounddown of D, and K the canonical divisor of X. We show that any smooth family of canonically polarized varieties over X\S is isotrivial if…