Ekaterina Amerik
Let $X$ be a complex projective bundle. We prove that $X$ admits an endomorphism of degree $>1$ and commuting with the projection to the base, if and only if $X$ trivializes after a finite covering. When $X$ is the projectivization of a vector bundle $E$ of rank 2, we prove that it has an endomorphism of degree $>1$ on a general fiber only if $E$ splits after a finite base change.
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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.
In this paper we give some properties of the algebraic and geometric structure of the endomorphisms monoid of a homogeneous vector bundle.
Let $B$ be a simply-connected projective variety such that the first cohomology groups of all line bundles on $B$ are zero. Let $E$ be a vector bundle over $B$ and $X={\mathbb P} (E)$. It is easily seen that a power of any endomorphism of $X$ takes fibers to fibers. We prove that if $X$ admits an endomorphism which is of degree greater than one on the fibers then $E$ splits into a direct sum of line bundles.
On Endomorphisms of Projective Algebraic Varieties
本文研究了射影簇内射映射的代数动力学，通过刻画迭代像、Stein分解和稳定性现象，完成Brion的结果。
Dynamics of Endomorphisms for Projective Bundles on Elliptic Curves
本文通过丛的转移函数研究椭圆曲线上射影丛满射自同态的动力学，证明了其Kawaguchi--Silverman猜想，并将结果推广至具有限生成Mori锥的光滑射影簇上的射影分裂丛。
Totally invariant divisors of endomorphisms of the projective space are expected to be always unions of linear spaces. Using logarithmic differentials we establish a lower bound for the degree of the non-normal locus of a totally invariant divisor. As a consequence we prove the linearity of totally invariant divisors for $\mathbb P^3$.
let f be an endomorphism of a complex projective space, of degree bigger than one. Let us call an algebraic subset exceptional for f, if its inverse image is set-theoretically equal to itself. J.-Y. Briend, S. Cantat and M. Shishikura proved that an irreducible set like this is a linear subspace. In this paper, we obtain a bound for the number of codimension-two exceptional subspaces. In particular, we show that an endomorphism of the projective plane can be completely ramified at nine points at most.
We first show that any connected algebraic group over a perfect field is the neutral component of the automorphism group scheme of some normal projective variety. Then we show that very few connected algebraic semigroups can be realized as endomorphisms of some projective variety X, by describing the structure of all connected subsemigroup schemes of End(X).
We study complex projective manifolds X that admit surjective endomorphisms f:X->X of degree at least two. In case f is etale, we prove structure theorems that describe X. In particular, a rather detailed description is given if X is a uniruled threefold. As to the ramified case, we first prove a general theorem stating that the vector bundle associated to a Galois covering of projective manifolds is ample (resp. nef) under very mild conditions. This is applied to the study of ramified endomorphisms of Fano manifolds with second Betti number one. It is conjectured that the projective space is …
Let X be a smooth irreducible complex projective curve of genus g > 1. In this paper, we give necessary and sufficient conditions for an unstable bundle of HN-lenght 2 to have a particular algebra of endomorphisms. Then, fixing the dimension of the algebra of endomorphisms we obtain a stratification of the moduli scheme such that each strata is a coarse moduli space. A particular case of interest is when the unstable bundles are simple. In that case the moduli space is fine. Topological properties of those moduli spaces are described and will depend on the generality of the curve X. Such resul…
Dynamics of Projectivized Toric Vector Bundles
该研究聚焦环面簇上射影丛的满自同态动力学，通过结构定理、动力次数控制分析及显式转移函数法，完成分类、验证猜想并证明相关丛无非自同构交换满自同态，推进代数几何动力学研究。
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…
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 $L$.
Fano manifolds of Picard number one whose co-tangent bundle is algebraically completely integrable system and its endomorphisms
本文针对皮卡数为一的射影Fano流形，在其余切丛为代数完全可积系统且切丛非nef的条件下证明了非常数自同态必为同构的猜想，并构造了满足余切丛可积条件的射影Fano流形例子，从而为三个已有主要结果提供替代证明。
It is conjectured that a Fano manifold of Picard number 1 which is not a projective space admits no endomorphisms of degree bigger than 1. Beauville confirmed this for hypersurfaces of projective space. We study this problem for hypersurfaces of an arbitrary Fano manifold of Picard number 1.
Etale endomorphisms of complex projective manifolds are constructed from two building blocks up to isomorphism if the good minimal model conjecture is true. They are the endomorphisms of abelian varieties and the nearly etale rational endomorphisms of weak Calabi-Yau varieties.
Birational conjugacies between endomorphisms on the projective plane
本文证明了特征0下射影平面上次数至少为2的支配自同态若双有理共轭则必自同构共轭，并给出正特征反例。
We prove the existence of global minimal models for rational morphisms $ϕ:{\mathbb P}^N\rightarrow{\mathbb P}^N$ of projective space defined over the field of fractions of a principal ideal domain.
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 of f^{-1}-stable prime divisors on (not necessarily smooth) projective varieties.
It is expected that a totally invariant divisor of a non-isomorphic endomorphism of the complex projective space is a union of hyperplanes. In this paper, we compute an upper bound for the degree of such a divisor. As a consequence, we prove the linearity of totally invariant divisors with isolated singularities.
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 $Aut(X)$…
We collect some results on endomorphisms on projective varieties related with the Kawaguchi-Silverman conjecture. We discuss certain condition on automorphism groups of projective varieties and positivity conditions on leading real eigendivisors of self-morphisms. We prove Kawaguchi-Silverman conjecture for endomorphisms on projective bundles on a smooth Fano variety of Picard number one. In the last section, we discuss endomorphisms and augmented base loci of their eigendivisors.
We characterize possible periodic subvarieties for surjective endomorphisms of complex abelian varieties in terms of the eigenvalues of the cohomological actions induced by the endomorphisms, extending previous work in this direction by Pink and Roessler. By applying our characterization to induced endomorphisms on Albanese varieties, we draw conclusions about the dynamics of surjective endomorphisms for a broad class of projective varieties. We also analyze several classes of surjective endomorphisms that are distinguished by properties of their cohomological actions.
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.
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.
Tensorial Constraints for Commuting Endomorphisms of the Generalized Tangent Bundle
本文研究一般切丛上交换内射的张量约束，扩展广义Kähler结构概念至非广义几乎复结构情形，通过Gröbner基技术构建并研究理想生成元。
Let X be a geometrically irreducible smooth projective curve over a field k. We describe the algebra of endomorphisms of indecomposable unstable vector bundles over X of rank 2 and degree d. Fixing some numerical invariants, namely the Harder-Narasimhan type and the dimension of the algebra of endomorphisms, we construct algebraic stacks and moduli schemes for such bundles.
We prove that the jacobian of a hyperelliptic curve y^2=f(x) is absolutely simple if deg(f)=q+1 where q is a power prime congruent to 5 modulo 8, the polynomial f(x) is irreducible over the ground field of characteristic zero and its Galois group is L_2(q).
We consider surjective endomorphisms f of degree > 1 on the projective n-space with n = 3, and f^{-1}-stable hypersurfaces V. We show that V is a hyperplane (i.e., deg(V) = 1) but with four possible exceptions; it is conjectured that deg(V) = 1 for any n > 1.
We prove that a smooth hypersurface of degree >2 and dimension >1 admits no endomorphism of degree >1 (for hyperquadrics this is due to Paranjape and Srinivas). We then collect some general results on endomorphisms of projective manifolds; we prove in particular that ramified endomorphisms occur only on varieties of negative Kodaira dimension, and we classify those Del Pezzo surfaces which admit such endomorphism.
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。
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 define an "ample canonical height" for an endomorphism on a projective variety, which is essentially a generalization of the canonical heights for polarized endomorphisms introduced by Call--Silverman. We formulate a dynamical analogue of the Northcott finiteness theorem for ample canonical heights as a conjecture, and prove it for endomorphisms on varieties of small Picard numbers, abelian varieties, and surfaces. As applications, for the endomorphisms which satisfy the conjecture, we show the non-density of the set of preperiodic points over a fixed number field, and obtain a dynamical Mo…
Isomorphisms and automorphisms of multiprojective bundles and symmetric powers of projective bundles
本文研究射影空间上多重射影丛及射影丛相对对称幂的同构分类问题，并进一步刻画了这些对象的自同构群。
In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$.
We study global deformations of certain projective bundles over projective spaces. We show that any projective global deformation of a projective bundle over $\bP^1$ carries the structure of a projective bundle over some projective space. Furthermore, we construct examples in arbitrary dimension $\ge 3$ of almost homogeneous Fano projective bundles over $\bP^2$ which can be globally deformed to non-almost homogeneous manifolds.
We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle H of a projective space bundle of rank r-1 over the projective line depends only on the the r-fold self-intersection of H . In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub-bundle with given r-fold self-intersection. Our proof exploits the unexpected property that every such complement comes equipped with the structure of a non trivial torsor under a vector bundle on the affine line with a double origin.
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…
Let $Φ$ be an endomorphism of $\SR(\bar{\Q})$, the projective line over the algebraic closure of $\Q$, of degree $\geq2$ defined over a number field $K$. Let $v$ be a non-archimedean valuation of $K$. We say that $Φ$ has critically good reduction at $v$ if any pair of distinct ramification points of $Φ$ do not collide under reduction modulo $v$ and the same holds for any pair of branch points. We say that $Φ$ has simple good reduction at $v$ if the map $Φ_v$, the reduction of $Φ$ modulo $v$, has the same degree of $Φ$. We prove that if $Φ$ has critically good reduction at $v$ and the reduction…
Let $f \colon X \to X$ be a surjective endomorphism of a normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure of $X$. Using this, we extend the second author's result to singular surfaces to the extent that either $X$ has an $f$-invariant non-constant rational function, or $f$ has infinitely many Zariski-dense forward orbits; this result is also extended to Adelic topology (which is finer than Zariski topology).
In this paper we prove the following theorem. Let $f$ be a dominant endomorphism of a smooth projective surface over an algebraically closed field of characteristic $0$. If there is no nonconstant invariant rational function under $f$, then there exists a closed point whose orbit under $f$ is Zariski dense. This result gives us a positive answer to the Zariski dense orbit conjecture proposed by Medvedev and Scanlon, by Amerik, Bogomolov and Rovinsky, and by Zhang, for endomorphisms of smooth projective surfaces. Moreover, we define a new canonical topology on varieties over an algebraically cl…
The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces.
The restricted Hitchin map of wobbly vector bundles
研究光滑射影曲线\(C\)上稳定秩\(2\)向量丛\(V\)的受限希钦映射\(h_V\)，通过分析一般摆动向量丛的幂零扭曲自同态，证明\(h_V\)一般有限并计算其度数，还得出在特定条件下\(h_V\)像的相关结论。
Isomorphism of Multiprojective Bundles and Projective Towers
本文研究不同维数光滑射影簇上射影丛的同构问题，证明不同射影空间上的多射影丛除平凡情形外不同构，并给出高度3射影塔顶层簇同构的充要条件。
We show that smooth projective horospherical varieties with nef tangent bundles are rational homogeneous spaces.
An almost Fano bundle is a vector bundle on a smooth projective variety that its projectivization is an almost Fano variety. In this paper, we prove that almost Fano bundles exist only on almost Fano manifolds and study rank 2 almost Fano bundles over projective spaces.
In previous papers it was shown that the left and right O-module structure of the jet bundles on the projective line differed. In this paper we show that similar statements hold for jet bundles on projective space in any dimension. We also consider some classes associated to jet bundles in the algebraic K-theory of projective space.
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。
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 fibrations. (For endomorphisms of projective spaces, see version 1).
Smooth blow up structures on projective bundles
本文在Hartshorne猜想等假设下分类了射影空间上具有光滑爆破结构的射影丛，验证了特定向量丛的相关性质，并得到了同时具有两种结构的簇的新例子。
We study ample vector bundles on smooth projective stacks. In particular, we prove that the tangent bundle for the weighted projective stack $\mathbb{P}(a_0,...,a_n)$ is ample. A result of Mori shows that the only smooth projective varieties with ample vector bundle are isomorphic to $\mathbb{P}^N$ for some $N$. Extending our geometric spaces from varieties to projective stacks, we are able to provide a new example. This leaves the open question of knowing if the only smooth projective stacks are the weighted projective stacks.
This short paper shows that a projective bundle over a smooth variety with a section can be uniquely recovered from the embedding of the section in its second infinitesimal neighborhood within the bundle.
On Serrano's conjecture on Projective bundles
本文针对高维光滑射影簇上的射影丛，考察严格 nef 除子情形下的 Serrano 猜想。
Log Calabi-Yau structure of projective threefolds admitting polarized endomorphisms
针对Broustet和Gongyo提出的容许极化自同态的正规射影簇为Calabi-Yau型的猜想，研究者基于等变极小模型纲领与典范丛公式建立框架，证明了光滑射影三维簇情形下该猜想成立。
In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite etale covering.
Suppose K is a field of characteristic 0, $K_a$ is its algebraic closure, p is an odd prime. Suppose, $f(x) \in K[x]$ is a polynomial of degree $n \ge 5$ without multiple roots. Let us consider a curve $C: y^p=f(x)$ and its jacobian J(C). It is known that the ring End(J(C)) of all $K_a$-endomorphisms of J(C) contains the ring $Z[ζ_p]$ of integers in the pth cyclotomic field (generated by obvious automorphisms of C). We prove that $End(J(C))=Z[ζ_p]$ if the Galois group of f over K is either the symmetric group $S_n$ or the alternating group $A_n$.
For a dominant rational self-map on a smooth projective variety defined over a number field, Kawaguchi and Silverman conjectured that the (first) dynamical degree is equal to the arithmetic degree at a rational point whose forward orbit is well-defined and Zariski dense. We prove this conjecture for surjective endomorphisms on smooth projective surfaces. For surjective endomorphisms on any smooth projective varieties, we show the existence of rational points whose arithmetic degrees are equal to the dynamical degree. Moreover, we prove that there exists a Zariski dense set of rational points h…
We give several mild conditions on a toric bundle on a nonsingular toric variety under which the projectivization of the toric bundle is Frobenius split.
The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including Calabi Yau m-folds and Del Pezzo surfaces, are indicated.
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