Zhong Guolei

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• Doctor of Philosophy
• PostDoc Position at Institute for Basic Science

Let X be a smooth projective variety of dimension $n\geq 2$ and $G\cong \mathbf {Z}^{n-1}$ a free abelian group of automorphisms of X over $\overline {\mathbf {Q}}$ . Suppose that G is of positive entropy. We construct a canonical height function $\widehat {h}_G$ associated with G , corresponding to a nef and big $\mathbf {R}$ -divisor, satisfying...

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f:X\rightarrow Y$$\end{document} be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a gene...

Let $X$ be a $\mathbb{Q}$-factorial klt projective variety admitting an int-amplified endomorphism $f$, i.e., the modulus of any eigenvalue of $f^*|_{\text{NS}(X)}$ is greater than $1$. We prove Kawaguchi-Silverman conjecture for $f$ and also any other surjective endomorphism of $X$: the first dynamical degree equals the arithmetic degree of any po...

Let X be a normal projective variety with klt singularities and LX a strictly nef ℚ-divisor on X. In this paper, we study the singular version of Serrano’s conjecture, i.e. the ampleness of KX + tLX for t ≫ 1. We show that, if X is a ℚ-factorial Gorenstein terminal threefold, then KX + tLX is ample for t ≫ 1 unless X is a weak Calabi–Yau variety (i...

Let be a projective klt pair, and a fibration to a smooth projective variety with strictly nef relative anti‐log canonical divisor . We prove that is a locally trivial fibration with rationally connected fibres, and the base is a canonically polarized hyperbolic manifold. In particular, when is a single point, we establish that is rationally connec...

Let \(f\colon X\to X\) be a dominant meromorphic self-map of a compact complex variety \(X\) in the Fujiki class \(\mathcal{C}\). If the topological degree of \(f\) is strictly larger than the other dynamical degrees of \(f\), we show that the number of isolated \(f\)-periodic points grows exponentially fast similarly to the topological degrees of...

Let $X$ be a smooth projective variety of dimension $n\geq 2$ and $G\cong\mathbf{Z}^{n-1}$ a free abelian group of automorphisms of $X$ over $\overline{\mathbf{Q}}$. Suppose that $G$ is of positive entropy. We construct a canonical height function $\widehat{h}_G$ associated with $G$, corresponding to a nef and big $\mathbf{R}$-divisor, satisfying t...

Let f : X → Y be a surjective morphism of Fano manifolds of Picard number 1 whose VMRTs at a general point are not dual defective. Suppose that the tangent bundle T X is big. We show that f is an isomorphism unless Y is a projective space. As applications, we study the bigness of the tangent bundles of complete intersections, del Pezzo manifolds, a...

In this paper, we extend the structure theorem for smooth projective varieties with nef tangent bundle to projective klt varieties whose tangent sheaf is either positively curved or almost nef. Specifically, we show that such a variety $X$, up to a finite quasi-\'etale cover, admits a rationally connected fibration $X \to A$ onto an abelian variety...

Let X be a Q$\mathbb {Q}$‐factorial compact Kähler klt threefold admitting an action of a free abelian group G, which is of positive entropy and of maximal rank. After running the G‐equivariant log minimal model program, we show that such X is either rationally connected or bimeromorphic to a Q‐complex torus. In particular, we fix an issue in the p...

Let $S$ be a non-uniruled (i.e., non-birationally ruled) smooth projective surface. We show that the tangent bundle $T_S$ is pseudo-effective if and only if the canonical divisor $K_S$ is nef and the second Chern class vanishes, i.e., $c_2(S)=0$. Moreover, we study the blow-up of a non-rational ruled surface with pseudo-effective tangent bundle.

Let $f:Y\to X$ be a finite morphism between Fano manifolds $Y$ and $X$ such that the Fano index of $X$ is greater than 1. On the one hand, when both $X$ and $Y$ are fourfolds of Picard number 1, we show that the degree of $f$ is bounded in terms of $X$ and $Y$ unless $X\cong\mathbb{P}^4$; hence, such $X$ does not admit any non-isomorphic surjective...

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.

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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\od...

Let $X$ be a $\mathbb{Q}$-factorial compact K\"ahler klt threefold admitting an action of a free abelian group $G$, which is of positive entropy and of maximal rank. After running the $G$-equivariant log minimal model program, we show that such $X$ is either rationally connected or bimeromorphic to a $Q$-complex torus. In particular, we fix an issu...

Let $X$ be a normal projective variety with only klt singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. In this paper, we study the singular version of Serrano's conjecture, i.e., the ampleness of $K_X+t L_X$ for sufficiently large $t\gg 1$. We show that, if $X$ is assumed to be a $\mathbb{Q}$-factorial Gorenstein terminal threefo...

Let $(X,\Delta)$ be a projective klt pair, and $f:X\to Y$ a fibration to a smooth projective variety $Y$ with strictly nef relative anti-log canonical divisor $-(K_{X/Y}+\Delta)$. We prove that $f$ is a locally constant fibration with rationally connected fibres, and the base $Y$ is a canonically polarized hyperbolic projective manifold. In particu...

Let $X$ be a normal projective threefold with mild singularities, and $L_X$ a strictly nef $\mathbb{Q}$-divisor on $X$. We first show the ampleness of $K_X+tL_X$ with sufficiently large $t$ if either the Kodaira dimension $\kappa(X)\neq 0$ or the augmented irregularity $q^{\circ}(X)\neq 0$. Second, we study the rational connectedness of a projectiv...

Let |$f:X\to X $| be a dominant self-morphism of an algebraic variety. Consider the set |$\Sigma _{f^{\infty }}$| of |$f$|-periodic subvarieties of small dynamical degree (SDD), the subset |$S_{f^{\infty }}$| of maximal elements in |$\Sigma _{f^{\infty }}$|⁠, and the subset |$S_f$| of |$f$|-invariant elements in |$S_{f^{\infty }}$|⁠. When |$X$| is...

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.

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 varie...

Let $f:X\to X $ be a dominant self-morphism of an algebraic variety over an algebraically closed field of characteristic zero. We consider the set $\Sigma_{f^{\infty}}$ of $f$-periodic (irreducible closed) subvarieties of small dynamical degree, the subset $S_{f^{\infty}}$ of maximal elements in $\Sigma_{f^{\infty}}$, and the subset $S_f$ of $f$-in...

Let $X$ be a rationally connected smooth projective variety of dimension $n$. We show that $X$ is a toric variety if and only if $X$ admits an int-amplified endomorphism with totally invariant ramification divisor. We also show that $X\cong (\mathbb{P}^1)^{\times n}$ if and only if $X$ admits a surjective endomorphism $f$ such that the eigenvalues...

In this short note, we study the normal compact K\"ahler threefold (possibly singular) $X$ admitting the action of an abelian group $G$ of maximal rank, all the non-trivial elements of which are of positive entropy. If such $X$ is further assumed to have at worst terminal singularities, then we prove that it is either a rationally connected project...

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 in Zhang (Adv Math 252:185–203, 2014, Theorem 1.3) for the case of polarized endomorphisms to the case of int-amplified endomorphisms. Assume further that X has a...

Let $X$ be a normal compact K\"ahler space of dimension $n$. A surjective endomorphism $f$ of such $X$ is int-amplified if $f^*\xi-\xi=\eta$ for some K\"ahler classes $\xi$ and $\eta$. First, we show that this definition generalizes the notation in the projective setting. Second, we prove that for the cases of $X$ being smooth, a surface or a three...

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 pr...