De-Qi Zhang

De-Qi Zhang
National University of Singapore | NUS · Department of Mathematics

PhD, Osaka University

About

118
Publications
6,083
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1,762
Citations
Citations since 2016
23 Research Items
780 Citations
2016201720182019202020212022050100150
2016201720182019202020212022050100150
2016201720182019202020212022050100150
2016201720182019202020212022050100150
Introduction
Skills and Expertise
Additional affiliations
September 1991 - present
National University of Singapore
Position
  • Professor (Full)
April 1989 - September 1991
Osaka University
Position
  • Faculty Member
October 1983 - March 1989
Osaka University, Japan
Position
  • MSc, PhD student

Publications

Publications (118)
Article
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We study zero entropy automorphisms of a compact Kähler manifold X. Our goal is to bring to light some new structures of the action on the cohomology of X, in terms of the so-called dynamical filtrations on \(H^{1,1}(X,{{\mathbb {R}}})\). Based on these filtrations, we obtain the first general upper bound on the polynomial growth of the iterations...
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Let X be a projective variety and σ a wild automorphism on X, i.e., whenever σ(Z)=Z for a non-empty Zariski-closed subset Z of X, we have Z=X. Then X is conjectured to be an abelian variety with σ of zero entropy (and proved to be so when dim⁡X≤2) by Z. Reichstein, D. Rogalski and J. J. Zhang in their study of projectively simple rings. This conjec...
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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...
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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 dens...
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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...
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Let $X$ be a quasi-projective variety and $f\colon X\to X$ a finite surjective endomorphism. We consider Zariski Dense Orbit Conjecture (ZDO), and Adelic Zariski Dense Orbit Conjecture (AZO). We consider also Kawaguchi-Silverman Conjecture (KSC) asserting that the (first) dynamical degree $d_1(f)$ of $f$ equals the arithmetic degree $\alpha_f(P)$ a...
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Twisted homogeneous coordinate rings are natural invariants associated to a projective variety X with an automorphism f. We study the Gelfand-Kirillov dimensions of these noncommutative algebras from the perspective of complex dynamics, by noticing that when X is a smooth complex projective variety, they essentially coincide with the polynomial log...
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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 the result of Prokhorov...
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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 $q > 1$. When $f$ is separable and $X$ is $Q$-Gorenstein and normal, we show that the anti-canonical divisor $-K_X$ i...
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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...
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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...
Preprint
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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 $f$-i...
Preprint
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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...
Preprint
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Let $X$ be a projective variety and $\sigma$ a wild automorphism on $X$, i.e., whenever $\sigma(Z) = Z$ for a Zariski-closed subset $Z$ of $X$, we have $Z = X$. Then $X$ is conjectured to be an abelian variety (and proved to be so when $\dim X \le 2$) by Z. Reichstein, D. Rogalski and J. J. Zhang. This conjecture has been generally open for more th...
Preprint
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In this paper we study dominant rational maps from a very general hypersurface $X$ of degree at least $n+3$ in the projective $(n+1)$-space ${\mathbb P}^{n+1}$ to smooth projective $n$-folds $Y$. Based on Lefschetz theory, Hodge theory, and Cayley-Bacharach property, we prove that there is no dominant rational map from $X$ to $Y$ unless $Y$ is unir...
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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 Q-factorial Kawamata log terminal projective variety admitting an int-am...
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Let $X$ be a normal projective variety and $f:X \to X$ a 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 characterizati...
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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 singl...
Preprint
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Let X be a compact Kahler manifold of dimension n. Let G be a group of zero entropy automorphisms of X. Let G0 be the set of elements in G which are isotopic to the identity. We prove that after replacing G by a suitable finite-index subgroup, G/G0 is a unipotent group of derived length at most n-1. This is a corollary of an optimal upper bound of...
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An endomorphism $f$ of a normal projective variety $X$ is polarized if $f^\ast H \sim qH$ for some ample Cartier divisor $H$ and integer $q > 1$. We first assert 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 $Y$...
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We determine positive-dimensional $G$-periodic proper subvarieties of an $n$-dimensional projective variety under the action of an abelian group of maximal rank $n-1$ and of positive entropy. The motivation of the paper is to understand the obstruction for $X$ to be $G$-equivariant birational to the quotient variety of an abelian variety modulo the...
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Let $V$ be a complex algebraic variety, homogeneous under the action of a complex algebraic group. We show that the log Kodaira dimension of $V$ is non-negative if and only if $V$ is a semi-abelian variety.
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Let $X$ be a projective surface or a hyperk\"ahler manifold and $G \le Aut(X)$. We give a necessary and sufficient condition for the existence of a non-trivial $G$-equivariant fibration on $X$. We also show that two automorphisms $g_i$ of positive entropy and polarized by the same nef divisor are the same up to powers, provided that either $X$ is n...
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A century ago, Camille Jordan proved that complex general linear group $GL_n(C)$ has the Jordan property: there is a Jordan constant $C_n$ such that every finite subgroup $H \subseteq GL_n(C)$ has an abelian subgroup $H_1$ of index $[H : H_1] \le C_n$. We show that every connected algebraic group $G$ (which is not necessarily linear) has the Jordan...
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Let G be a connected linear algebraic group over an algebraically closed field k, and let H be a connected closed subgroup of G. We prove that the homogeneous variety G/H is a rational variety over k whenever H is solvable, or when dim(G/H) < 11 and characteristic(k) = 0. When H is of maximal rank in G, we also prove that G/H is rational if the max...
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Let G be a group of automorphisms of a compact K\"ahler manifold X of dimension n and N(G) the subset of null-entropy elements. Suppose G admits no non-abelian free subgroup. Improving the known Tits alternative, we obtain that, up to replace G by a finite-index subgroup, either G/N(G) is a free abelian group of rank < n-1, or G/N(G) is a free abel...
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Let X be a normal projective variety of dimension n > 2 admitting the action of the group G := Z^{n-1} such that every non-trivial element of G is of positive entropy. We show: `X is not rationally connected' ==> `X is G-equivariant birational to the quotient of a complex torus' <==> `K_X + D is pseudo-effective for some G-periodic effective fracti...
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For every smooth complex projective variety $W$ of dimension $d$ and nonnegative Kodaira dimension, we show the existence of a universal constant $m$ depending only on $d$ and two natural invariants of the general fibres of an Iitaka fibration of $W$ such that the pluricanonical system $|mK_W|$ defines an Iitaka fibration. This is a consequence of...
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We classify normal supersingular K3 surfaces Y with total Milnor number 20 in characteristic p, where p is an odd prime that does not divide the discriminant of the Dynkin type of the rational double points on Y .
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Let X be a projective variety which is algebraic Lang hyperbolic. We show that Lang's conjecture holds (one direction only): X and all its subvarieties are of general type and the canonical divisor K_X is ample at smooth points and Kawamata log terminal points of X, provided that K_X is Q-Cartier, no Calabi-Yau variety is algebraic Lang hyperbolic...
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We show that a compact Kähler manifold $X$ is a complex torus if both the continuous part and discrete part of some automorphism group $G$ of $X$ are infinite groups, unless $X$ is bimeromorphic to a non-trivial $G$ -equivariant fibration. Some applications to dynamics are given.
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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.
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We survey our recent papers (some being joint ones) about the relation between the geometry of a compact K\"ahler manifold and the existence of automorphisms of positive entropy on it. We also use the language of log minimal model program (LMMP) in biraitonal geometry, but not its more sophisticated technical part. We give applications of LMMP to p...
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We slightly extend a result of Oguiso on birational or automorphism groups (resp. of Lazi\'c - Peternell on Morrison-Kawamata cone conjecture) from Calabi-Yau manifolds of Picard number two to arbitrary singular varieties X (resp. to klt Calabi-Yau pairs in broad sense) of Picard number two. When X has only klt singularities and is not a complex to...
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For an automorphism group G on an n-dimensional (n > 2) normal projective variety or a compact K\"ahler manifold X so that G modulo its subgroup N(G) of null entropy elements is an abelian group of maximal rank n-1, we show that N(G) is virtually contained in Aut_0(X), the X is a quotient of a complex torus T and G is mostly descended from the symm...
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Let X be a complex projective variety and D a reduced divisor on X. Under a natural minimal condition on the singularities of the pair (X, D), which includes the case of smooth X with simple normal crossing D, we ask for geometric criteria guaranteeing various positivity conditions for the log-canonical divisor K_X+D. By adjunction and running the...
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We prove two results about the natural representation of a group G of automorphisms of a normal projective threefold X on its second cohomology. We show that if X is minimal then G, modulo a normal subgroup of null entropy, is embedded as a Zariski-dense subset in a semi-simple real linear algebraic group of real rank < 3. Next, we show that X is a...
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We describe a bound on the degree of the generators for some adjoint rings on surfaces and threefolds.
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We prove that upto isomorphisms there are exactly, three rational log Enriques surface of index four and Type A17, and three rational log Enriques surfaces of index two and actual Type A17 + A1.
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We confirm, to some extent, the belief that a projective variety X has the largest number (relative to the dimension of X) of independent commuting automorphisms of positive entropy only when X is birational to a complex torus or a quotient of a torus. We also include an addendum to an early paper though it is not used in the present paper.
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We use a concise method to construct pseudo-automorphisms f_n of positive entropy on the blowups of the projective n-space for all n > 1 and more generally on the blowups of products of projective spaces. These f_n for n > 3 and those on the blowups of products of projective spaces seem to be the first examples of pseudo-automorphisms of positive e...
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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.
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We determine the geometric structure of a minimal projective threefold having two `independent and commutative' automorphisms of positive topological entropy, and generalize this result to higher-dimensional smooth minimal pairs (X, G). As a consequence, we give an effective lower bound for the first dynamical degree of these automorphisms of X fit...
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We prove a structure theorem for non-isomorphic endomorphisms of weak Q-Fano threefolds, or more generally for threefolds with big anti-canonical divisor. Also provided is a criterion for a fibred rationally connected threefold to be rational. As a consequence, we show (without using the classification) that every smooth Fano threefold having a non...
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For a compact Kähler manifold X and a strongly primitive automorphism g of positive entropy, it is shown that X has at most ρ(X) of g-periodic prime divisors. When X is a projective threefold, every prime divisor containing infinitely many g-periodic curves, is shown to be g-periodic (a result in the spirit of the Dynamic Manin–Mumford conjecture a...
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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.
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For a compact hyperk\"ahler manifold X, we show certain Zariski decomposition for every pseudo-effective R-divisor, and give a sufficient condition for X to be bimeromorphic to a (holomorphic) Lagrangian fibration. We also prove that any sequence of D-flops between projective hyperk\"ahler manifolds terminates after finite steps.
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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.
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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...
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We show that the dynamics of automorphisms on all projective complex manifolds X (of dimension 3, or of any dimension but assuming the Good Minimal Model Program or Mori's Program) are canonically built up from the dynamics on just three types of projective complex manifolds: complex tori, weak Calabi-Yau manifolds and rationally connected manifold...
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We show that if a compact Kahler manifold X admits a cohomologically hyperbolic surjective endomorphism then its Kodaira dimension is non-positive. This gives an affirmative answer to a conjecture of Guedj in the holomorphic case. The main part of the paper is to determine the geometric structure and the fundamental groups (up to finite index) for...
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We prove a theorem of Tits type about automorphism groups for compact Kähler manifolds, which has been conjectured in the paper [9].
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We classify surjective self-maps (of degree at least two) of affine surfaces according to the log Kodaira dimension.
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We give an algebro-geometric approach towards the dynamics of automorphisms/endomorphisms of projective varieties or compact K\"ahler manifolds, try to determine the building blocks of automorphisms /endomorphisms, and show the relation between the dynamics of automorphisms/endomorphisms and the geometry of the underlying manifolds.
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We show that every supersingular K3 surface in characteristic 2 with Artin invariant less than or equal to 2 is obtained by the Kummer type construction of Schroeer.
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For every n-dimensional projective manifold X of Kodaira dimension 2 we show that Φ |MKX | is birational to an Iitaka fibration for a computable positive integer M = M(b, Bn−2), where b> 0 is minimal with |bKF | ̸ = ∅ for a general fibre F of an Iitaka fibration of X, and where Bn−2 is the Betti number of a smooth model of the canonical Z/(b)-cover...
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We show the existence of an anti-pluricanonical curve on every smooth projective rational surface X which has an infinite group G of automorphisms of either null entropy or of type Z . Z (semi-direct product), provided that the pair (X, G) is minimal. This was conjectured by Curtis T. McMullen (2005) and further traced back to Marat Gizatullin and...
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In this article we present a 3-dimensional analogue of a well-known theorem of Bombieri (Inst Hautes Etudes Sci Publ Math 42:171–219, 1973) which characterizes the bi-canonical birationality of surfaces of general type. Let X be a projective minimal threefold of general type with \({\mathbb{Q}}\)-factorial terminal singularities and the geometric g...
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First, we shall formulate and prove Theorem of Lie-Kolchin type for a cone and derive some algebro-geometric consequences. Next, inspired by a recent result of Dinh and Sibony we pose a conjecture of Tits type for a group of automorphisms of a complex variety and verify its weaker version. Finally, applying Theorem of Lie-Kolchin type for a cone, w...
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We shall determine the uniquely existing extension of the alternating group of degree 6 (being normal in the group) by a cyclic group of order 4, which can act on a complex K3 surface.
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We give an effective upper bound of |Bir(X)| for the birational automorphism group of an irregular n-fold (with n = 3) of general type in terms of the volume V = V(X) under an “albanese smoothness and simplicity” condition. To be precise, \(|{\rm Bir}(X)| \leq d_3 V^{10}\) . An optimum linear bound \(|{\rm Bir}(X)| \leq \frac{1}{3} \times 42^3 V\)...
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In this note, we consider all possible extensions G of a non-trivial perfect group H acting faithfully on a K3 surface X. The pair (X,G) is proved to be uniquely determined by G if the transcendental value of G is maximum. In particular, we have , if H is the alternating group A5 and normal in G.
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Let $X$ be a projective minimal Gorenstein 3-fold of general type with canonical singularities. We prove that the 5-canonical map is birational onto its image.
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We prove a non-vanishing theorem of the cohomology H0 of the adjoint divisor KX+⌈L⌉ where ⌈L⌉ is the round up of a nef and big Q-divisor L.
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Let X be a smooth projective minimal 3-fold of general type. We prove the sharp inequality K^3_X >= (2 /3)(2p_g(X) - 5), an analogue of the classical Noether inequality for algebraic surfaces of general type
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In this note, we report some progress we made recently on the automorphisms groups of K3 surfaces. A short and straightforward proof of the impossibility of Z/(60) acting purely non-symplectically on a K3 surface, is also given, by using Lefschetz fixed point formula for vector bundles.
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We prove a non-vanishing theorem of the cohomology H^0 of the adjoint divisor K_X + L' where L' is the round up of a nef and big Q-divisor L.
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The alternating group of degree 6 is located at the junction of three series of simple non-commutative groups: simple sporadic groups, alternating groups and simple groups of Lie type. It plays a very special role in the theory of finite groups. We shall study its new roles both in a finite geometry of a certain pentagon in the Leech lattice and al...
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We show that normal K3 surfaces with ten cusps exist in and only in characteristic 3. We determine these K3 surfaces according to the degrees of the polarizations. Explicit examples are given.
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In this note, we consider K3 surfaces X with an action by the alternating group A_5. We show that if a cyclic extension A_5 . C_n acts on X then n = 1, 2, or 4. We also determine the A_5-invariant sublattice of the K3 lattice and its discriminant form.
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We classify equivariantly Gorenstein $\log$ del Pezzo surfaces with boundaries at infinity and with finite group actions such that the quotient surface modulo the finite group has Picard number one. We also determine the corresponding finite groups.
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We classify equivariantly Gorenstein log del Pezzo surfaces with boundaries at infinity and with finite group actions such that the quotient surface modulo the finite group has Picard number one. We also determine the corresponding finite groups. Better figures are available upon request.
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We express explicitly the integral closures of some ring extensions; this is done for all Bring-Jerrard extensions of any degree as well as for all general extensions of degree < 6; so far such an explicit expression is known only for degree < 4 extensions. As a geometric application we present explicitly the structure sheaf of every Bring-Jerrard...
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In these notes, we consider self-maps of degree > 1 on a weak del Pezzo surface X of degree < 8. We show that there are exactly 12 such X, modulo isomorphism. In particular, K_X^2 > 2, and if X has one self-map of degree > 1 then for every positive integer d there is a self-map of degree d^2 on X. We prove the Sato conjecture in the present case, t...
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We survey some recent progress in the study of algebraic varieties X with log terminal singularities, especially, the uni-ruledness of the smooth locus X^0 of X, the fundamental group of X^0 and the automorphisms group on (smooth or singular) X when dim X = 2. The full automorphism groups of a few interesting types of K3 surfaces are described, mai...
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We investigate when the fundamental group of the smooth part of a K3 surface or Enriques surface with Du Val singularities, is finite. As a corollary we give an effective upper bound for the order of the fundamental group of the smooth part of a certain Fano 3-fold. This result supports Conjecture A below, while Conjecture A (or alternatively the r...
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In this note we shall determine all actions of groups of prime order p with p > 3 on Gorenstein del Pezzo (singular) surfaces Y of Picard number 1. We show that every order-p element in Aut(Y) (= Aut(Y'), Y' being the minimal resolution of Y) is lifted from a projective transformation of the projective plane. We also determine when Aut(Y) is finite...
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In this paper we prove that a normal Gorenstein surface dominated by $\proj^2$ is isomorphic to a quotient $\proj^2/G$, where $G$ is a finite group of automorphisms of $\proj^2$ (except possibly for one surface $V_8'$). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.
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In this paper we prove that a normal Gorenstein surface dominated by the projective plane P^2 is isomorphic to a quotient P^2/G, where G is a finite group of automorphisms of P^2 (except possibly for one surface V_8'). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.
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The aim of this note is to characterize a K3 surface of Klein-Mukai type in terms of its symmetry.
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We classify minimal pairs (X,G) of a smooth rational projective surface X and a finite group G of automorphisms on X. We also determine the fixed locus X G and the quotient surface Y=X/G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Mori’s extremal ray theory and recent results of V. A. Alexeev...
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We classify minimal pairs (X, G) for smooth rational projective surface X and finite group G of automorphisms on X. We also determine the fixed locus X^G and the quotient surface Y = X/G as well as the fundamental group of the smooth part of Y. The realization of each pair is included. Mori's extremal ray theory and recent results of Alexeev and al...
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We present a complete list of extremal elliptic K3 surfaces. There are altogether 325 of them. The first 112 coincides with Miranda-Persson's list for semi-stable ones. The data include the transcendental lattice which determines uniquely the K3 surface by a result of Shioda and Inose, the singular fibre type and the Mordell Weil group. As an appli...
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We shall give a proof for Vorontsov's Theorem and apply this to classify log Enriques surfaces with large prime canonical index.
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A smooth rational surface X is a Coble surface if the anti-canonical linear system is empty while the anti-bicanonical linear system is non-empty. In this note we shall classify these X and consider the finiteness problem of the number of negative curves on X modulo automorphisms.
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In the present paper we describe the K3 surfaces admitting order 11 automorphisms and apply this to classify log Enriques surfaces of global index 11.
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Let X be a K3 surface with the Neron-Severi lattice S_X and transcendental lattice T_X. Nukulin considered the kernel H_X of the natural representation Aut(X) ---> O(S_X) and proved that H_{X} is a finite cyclic group with phi(h(X))) | t(X) and acts faithfully on the space H^{2,0}(X) = C omega_{X}, where h(X) = ord(H_X), t(X) = rank T_X and phi(.)...
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Let X be a K3 surface with an involution g which has non-empty fixed locus X^g and acts non-trivially on a non-zero holomorphic 2-form. We shall construct all such pairs (X, g) in a canonical way, from some better known double coverings of log del Pezzo surfaces of index at most 2 or rational elliptic surfaces, and construct the only family of each...
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We show that there are exactly, up to isomorphisms, seven extremal log Enriques surfaces Z and construct all of them; among them types D_{19} and A_{19} have been shown of certain uniqueness by M. Reid. We also prove that the (degree 3 or 2) canonical covering of each of these seven Z has either X_3 or X_4 as its minimal resolution. Here X_3 (resp....
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In one of their early works, Miranda and Persson have classified all possible configurations of singular fibers for semistable extremal elliptic fibrations on K3 surfaces. They also obtained the Mordell-Weil groups in terms of the singular fibers except for 17 cases where the determination and the uniqueness of the groups were not settled. In this...
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We shall show that there is only one (resp. two) rational log Enriques surface(s) of Dynkin type D-eighteen (resp. A-eighteen).
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We prove that upto isomorphisms there are at least one and at most three rational log Enriques surfaces of index 3 and Type A17.
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LetSbe a rational projective algebraic surface, with at worst quotient singular points but with no rational double singular points, such thatIKS∼0 for some minimal positive integerI. IfI=2, we prove that the fundamental group π1(S−SingS) is soluble of order ≤256 (Theorem 1). IfI≥3 orShas at worst rational double singular points, then, in general, π...
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We shall characterize the unique K3 surface of discriminant 3 or 4, called the most algebraic K3 surfaces by Vinberg, in terms of the fixed locus of an automorphism on it. Based on this result, we show that there is, up to isomorphisms, only one rational log Eriques surface of Type D19 and one of Type A19. Introduction. We shall characterize the un...
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Let (S; ) be a log surface with at worst log canonical singularities and reduced boundary such that (KS + ) is nef and big. We shall prove that So = S SingS either has nite fundamental group or is ane-ruled. Moreover, 1(So) and the structure of S are determined in some sense when =0 .

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Is the automorphism group of algebraic varieties or compact complex varieties Jordan?