Amplified endomorphisms of Fano fourfolds
Jia Jia, Guolei Zhong
本文证明具有圆锥曲线丛结构的光滑 Fano 四维簇为环簇当且仅当它允许放大自同态，此时该簇为有理簇。
设 $X$ 为具有圆锥曲线丛结构的光滑 Fano 四维簇。我们证明 $X$ 是环簇当且仅当 $X$ 允许放大自同态；在这种情况下，$X$ 是有理簇。
Let $X$ be a smooth Fano fourfold admitting a conic bundle structure. We show that $X$ is toric if and only if $X$ admits an amplified endomorphism; in this case, $X$ is a rational variety.
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We prove Kawaguchi-Silverman conjecture for all surjective endomorphisms on every smooth rationally connected variety admitting an int-amplified endomorphism.
On smooth rationally connected projective threefolds of Picard number two admitting int-amplified endomorphisms
该研究证明Picard数为2的光滑有理连通射影三维簇为环面簇的充要条件是存在int-amplified自同态，并得出爆破等变下$\u202e\bb{P}^3$非同构满自同态的全不变光滑曲线必为直线的推论。
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.
Characterization of $Q$-complex tori via endomorphisms -- an addendum to "Int-amplified endomorphisms of compact Kähler spaces''
本文针对典范除子伪有效的正规紧 Kähler klt 空间，证明允许 int-amplified 自同态即可推出其为 $Q$-复环面，并据此得到单连通紧 Kähler 光滑三维簇的有理连通性结论。
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.
On endomorphisms of projective varieties with numerically trivial canonical divisors
本文研究具有数值平凡典范除子的klt射影簇的拟放大与放大自同态，证明经迭代和等变双有理收缩拟放大自同态可下降为放大自同态，给出超凯勒簇、阿贝尔簇上相关自同态的判别与周期点性质。
Let $X$ be a normal compact Kähler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^*ξ-ξ=η$ for some Kähler classes $ξ$ and $η$. First, we show that this definition generalizes the notion in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a threefold with mild singularities, if $X$ admits an int-amplified endomorphism with pseudo-effective canonical divisor, then it is a $Q$-torus. Finally, we consider a normal compact Kähler threefold $Y$ with only terminal singularities and show that, replacing $f$ by a pos…
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 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 $Aut(X)$…
In this paper, we obtain a characterization of toric varieties via int-amplified endomorphisms. We prove that if $f \colon X \to X$ is an int-amplified endomorphism of a smooth complex projective variety $X$, then $X$ is toric if and only if $f_*L$ is a direct sum of line bundles on $X$ for every line bundle $L$.
We investigate int-amplified endomorphisms on normal projective surfaces. We prove that the output of the equivariant MMP is either a Q-abelian surface, a (equivariant) quasi-étale quotient of a smooth projective surface, a Mori dream space, or a projective cone of an elliptic curve.
Construction of log Calabi--Yau boundaries from int-amplified endomorphisms
本文证明具有int-amplified自同态的$\mathbb{Q}$-Gorenstein射影复簇是Calabi--Yau型的，从而构造了log Calabi--Yau边界。
We consider the potential density of rational points on an algebraic variety defined over a number field $K$, i.e., the property that the set of rational points of $X$ becomes Zariski dense after a finite field extension of $K$. For a non-uniruled projective variety with an int-amplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an int-amplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we p…
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 an étale in codimension one finite morphism $\widetilde{X} \to X$ such that $\widetilde{X}$ is of Fano type over its albanese variety. As a corollary, if we further assume that $X$ is smooth and rationally connected, then $X$ is of Fano type.
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 for polarized endomorphisms of uniruled threefolds and fourfolds. As a consequence, we show that every smooth Fano threefold with a polarized endomorphism of degree > 1, is rational.
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 of int-amplified endomorphisms. Assume further that $X$ has at worst Kawamata log terminal singularities. We prove that the total number of $f^{-1}$-stable prime divisors has an optimal upper bound $\dim X+ρ(X)$, where $ρ(X)$ is the Picard number. Also, we give a sufficient condition for $X$ to be rationally connected and simply connected. Finally, by running…
Kawaguchi-Silverman conjecture for int-amplified endomorphism
本文在 $\mathbb{Q}$-因子的 klt 射影簇上，对内蕴放大自同态及任意满射自同态证明了 Kawaguchi-Silverman 猜想，将第一动力次数与 Zariski 稠密轨道点的算术次数相等这一结论从极化情形推广到更一般的内蕴放大情形。
Positivity of compact Kähler varieties admitting an int-amplified endomorphism
本文从切层正性角度研究允许int-amplified自同态的紧致Kähler簇，证明其切层弱正弯曲，并建立Kähler版本的Yoshikawa结构定理，同时证明极小模型存在性及自同态提升性质。
Images of toric variety and amplified endomorphism of weak Fano threefolds
证明Picard秩为2的弱Fano三维流形若为射影环面簇的像则是环面的，并证明具有放大自同态的此类流形也是环面的。
Numerical action for endomorphisms
本文研究射影簇满射自同态在除子数值群上的作用，证明上同调双曲性与拟放大性及放大性的等价关系，并引入锥体谱和生成正闭链两个新概念。
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 $\mathbb{P}^1$ 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.
Families of smooth Fano fourfolds of Picard rank 1 without Bott vanishing
本文证明当前已知的 Picard 秩 1 指数 1 的光滑 Fano 四重族满足 χ(X,T_X)<0，结合指数>1 情形，表明除射影空间外所有已知 Picard 秩 1 的光滑 Fano 四重均不满足 Bott 消失，从而任何具有大于 1 次自同态的此类簇必为 P^4。
Ulrich and Instanton Bundles on Special Cubic Fourfolds
本文研究特殊三次四维流形等射影超曲面上的瞬子丛与Ulrich丛，证明低秩低荷瞬子的非循环扩张可形变得到低秩Ulrich丛并约束流形所属Hassett除子，还考察了特定判别式的三次除子。
Endomorphisms of Fano 3-folds and log Bott vanishing
本文将射影簇内放大自同态的博特消失定理推广到对数版本，应用于法诺三维簇，重新证明并推广了光滑复法诺三维簇相关结果至特征p情形。
Boundedness of finite morphisms onto Fano manifolds with large Fano index
本文研究了到 Fano 指数大于 1 的 Fano 流形上的有限态射的有界性，证明了特定四维流形情形下态射次数的有界性及无非同构满射自同态，并在 int-amplified 自同态条件下证明了流形的 toric 性及奇异二次曲面的分类。
In this paper, we study the global $F$-splitting of varieties admitting an int-amplified endomoprhism. We prove that surfaces admitting an int-amplified endomorphism are of dense globally $F$-split type and, in particular, of Calabi-Yau type.
Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold (originally obtained by Claire Voisin and the first author in 2007).
Mumford constructed a family of abelian fourfolds with special stucture not characterized by endomorphism ring. Galluzzi showed that the weight 2 Hodge structure of such a variety decomposes into Hodge substructures via the action of Mumford-Tate group, one of which is of K3 type with Hodge number (1,7,1). We will compute the intersection form of such a Hodge structure and provide a canonical integral Hodge structure on this subspace. Furthermore, we shall show two applications of this invariant. Firstly, using the above formula we shall show that every K3 surface obtained in this way will adm…
Varieties Admitting Polarized Galois Endomorphisms
本文证明了允许内放大伽罗瓦自同态的光滑复射影簇的极大有理连通纤维化为环面纤维化，从而推出有理连通时该簇为环面簇，皮卡德数一时为射影空间，并验证了对数 Calabi-Yau 型猜想。
Rigidity of rationally connected smooth projective varieties from dynamical viewpoints
本文从动力系统视角研究了有理连通光滑射影簇的刚性，证明了其成为环面簇或$(\mathbb{P}^1)^{\times n}$的充要条件分别取决于其自同态的分支除子不变性与特征值性质。
De Rham-Betti Groups of Type IV Abelian Fourfolds
研究\(\bar{\mathbb{Q}}\)上几类阿贝尔簇的德拉姆 - 贝蒂群，通过采用不同于常规的方法，利用相关结果、分析及约束，确定了简单CM阿贝尔四维簇等的\(\mathrm{G}_{\mathrm{dRB}}(A)=\mathrm{MT}(A)\) 。
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