Yi Gu
For a relatively minimal surface fibration $f: X\to C$, the equivariant automorphism group of $f$ is, roughly speaking, the group of automorphisms of $X$ preserving the fibration structure. We present a classification of such fibrations of fibre genus $g\ge 1$ with smooth generic fibre over an arbitrary algebraically closed field $\mathbf{k}$ whose equivariant automorphism group is infinite.
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We study the order of automorphism groups of cyclic covering fibrations of a ruled surface. Arakawa and later Chen studied it for hyperelliptic fibrations and gave the upper bound. The purpose of present paper is to pursue the analog for cyclic covering fibrations.
We develop technics of birational geometry to study automorphisms of affine surfaces admitting many distinct rational fibrations, with a particular focus on the interactions between automorphisms and these fibrations. In particular, we associate to each surface $S$ of this type a graph encoding equivalence classes of rational fibrations from which it is possible to decide for instance if the automorphism group of $S$ is generated by automorphisms preserving these fibrations.
Bounds for the order of automorphism groups of cyclic covering fibrations of an elliptic surface
本文针对椭圆曲面的有限循环覆盖纤维化，利用纤维亏格、基曲线亏格、覆盖次数和相对典范除子平方，给出了自同构群有限子群阶的上界估计。
Automorphism groups of Mori Del Pezzo fibrations over an irrational curve
本文研究了Mori Del Pezzo纤维 over 一个正亏曲线的自同构群，并给出了Bir(C×P²)中最大连通代数子群的分类。
Jacobian elliptic fibrations on K3s with a non-symplectic automorphism of order 3
本文依据3阶非辛自同构对纤维的作用，结合Kneser–Nishiyama方法分类相应K3曲面上的Jacobian椭圆纤维化，并给出可约纤维类型与σ不变Weierstrass方程。
We analyze K3 surfaces admitting an elliptic fibration $E$ and a finite group $G$ of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration $E/G$ comparing its properties to the ones of $E$. We show that if $E$ admits an $n$-torsion section, its quotient by the group of automorphisms induced by this section admits again an $n$-torsion section. Considering automorphisms coming from the base of the fibration, we can describe the Mordell--Weil lattice of a fibration described by Kloosterman. We give the isometries between lattices described by th…
We study slopes of finite cyclic covering fibrations of a fibered surface. We give the best possible lower bound of the slope of these fibrations. We also give the slope equality of finite cyclic covering fibrations of a ruled surface and observe the local concentration of the global signature of these surfaces on a finite number of fiber germs. We also give an upper bound of the slope of finite cyclic covering fibrations of a ruled surface.
Nikulin has classified all finite abelian groups acting symplectically on a K3 surface and he has shown that the induced action on the K3 lattice $U^3\oplus E_8(-1)^2$ depends only on the group but not on the K3 surface. For all the groups in the list of Nikulin we compute the invariant sublattice and its orthogonal complement by using some special elliptic K3 surfaces.
We study the maximal Salem degree of automorphisms of K3 surfaces via elliptic fibrations. By generalizing \cite{EOY14}, we establish a characterization of such maximum in terms of elliptic fibrations with infinite automorphism groups. As an application, we show that any supersingular K3 surface in odd characteristic has an automorphism the entropy of which is the natural logarithm of a Salem number of degree $22$.
For a complex surface of general type with a relatively minimal genus 2 fibration, the bounds of the orders of the automorphism group of the fibration, of its abelian subgroups and of its cyclic subgroups are determined as linear functions of $c^2_1$. Most of them are the best.
Nishiyama introduced a lattice theoretic classification of the elliptic fibrations on a $K3$ surface. In a previous paper we used his method to exhibit $52$ elliptic fibrations, up to isomorphisms, of the singular $K3$ surface of discriminant $-12$. We prove here that the list is complete with a $53$th fibration, thanks to a remark of Elkies and Schütt. We characterize the fibration both theoretically and with a Weierstrass model.
Automorphism groups and Homogeneous fibrations on projective varieties with nef anticanonical divisors
本文研究具有 nef 反典范除子的代数簇的 Albanese 态射与极大有理链连通纤维化的齐次结构，构造自然群同态诱导 Aut^0 的 Chevalley 分解（相差等源），并类比 Nishi-Matsumura 定理，用现代语言重证之。
Nikulin and Vinberg proved that there are only a finite number of lattices of rank $\geq 3$ that are the Néron-Severi group of projective K3 surfaces with a finite automorphism group. The aim of this paper is to provide a more geometric description of such K3 surfaces $X$, when these surfaces have moreover no elliptic fibrations. In that case we show that such K3 surface is either a quartic with special hyperplane sections or a double cover of the plane branched over a smooth sextic curve which has special tangencies properties with some lines, conics or cuspidal cubic curves. We then study th…
We classify complex K3 surfaces of zero entropy admitting an elliptic fibration with only irreducible fibers. These surfaces are characterized by the fact that they admit a unique elliptic fibration with infinite automorphism group. We furnish an explicit list of 32 Néron-Severi lattices corresponding to such surfaces. Incidentally, we are able to decide which of these 32 classes of surfaces admit a unique genus 1 fibration. Finally, we prove that all K3 surfaces with Picard rank >=19 and infinite automorphism group have positive entropy.
On symplectic automorphisms of a surface with genus two fibration and their action on $\mathrm{CH}_0$
本文研究具有 genus 二纤维的表面的辛自同构，证明当χ(O_S)≥5时，自同构群大小不超过2，并验证辛自同构对CH_0的Albanese核作用为平凡。
We study the polynomial fibration induced by the equation of the Klein surfaces obtained as quotient of finite linear groups of automorphisms of the plane; this surfaces are of type A, D, E, corresponding to their singularities. The generic fibre of the polynomial fibration is a surface defined over the function field of the line. We proved that it is not rational in cases D, E, although it is obviously rational in the case A. The group of automorphisms of the Klein surfaces is also described, and is linear and of finite dimension in cases D, E; this result being obviously false in case A.
Equivariant automorphism group and real forms of complexity-one varieties
本文研究了复杂度为一的G-变种的等变自同构群的可代表性，并证明其在特定条件下仅存在有限多个实形式。
Let E be a toric fibration arising from symplectic reduction of a direct sum of line bundles over (almost-) Kähler base B. Then each torus-fixed point of the toric manifold fiber defines a section of the fibration. Let L_a be convex line bundles over B, A_a smooth divisors of B arising as the zero loci of generic sections of L_a, and \a:B\to E a particular fixed-point section of E. Further assume the \{A_a\} to be mutually disjoint. We compute genus-0 Gromov--Witten invariants of the blowup of E along \a(\coprod_a A_a) in terms of genus-0 Gromov--Witten invariants of B and of \{A_a\}, the matr…
Toric sheaves, stability and fibrations
本文研究了极化环面簇上等变反射层沿环面纤维化拉回的斜率稳定性，证明了在绝热极化下稳定性得以保持，给出了严格半稳定情形的充要条件，并展示了其在切层稳定扰动及模空间单射映射中的应用。
On two-elementary K3 surfaces with finite automorphism group
本研究针对皮卡秩11-13、带有限自同构群和二阶截面雅可比椭圆纤维化的复代数K3曲面，证明其存在射影四次超曲面双有理模型，构造椭圆纤维化标架，确定有理曲线对偶图等几何结构。
We consider K3 surfaces of Picard rank 14 which admit a purely nonsymplectic automorphism of order 16. The automorphism acts on the second cohomology group with integer coefficients and we compute the invariant sublattice for the action. We show that all of these K3 surfaces admit an elliptic fibration and we compute the invariant lattices in a geometric way by using special curves of the elliptic fibration. The computation of these lattices plays an important role when one wants to study moduli spaces and mirror symmetry for lattice polarized K3 surfaces.
Counting elliptic fibrations on K3 surfaces
本文研究任意复射影 K3 曲面上模自同构的 Jacobian 椭圆纤维化计数问题，给出求解方法并以显式示例加以说明。
The Automorphism Equivariant Hitchin Index
本文针对一维复环面下黎曼曲面自同构对秩2奇次固定行列式Higgs丛模空间上T等变行列式线丛上同调的作用，定义并研究自同构等变Hitchin指标，证明了其基于商曲面抛物Higgs丛模空间上同调配对的计算公式。
Let $X$ be a projective surface or a hyperkähler 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 not an abelian surface or the $g_i$ share at least one common periodic point. The surface case is known among experts, but we treat this case together with the hyperkähler case using the same language of hyperbolic lattice and following Ratcliffe or Ogu…
We calculate the automorphism group of certain Enriques surfaces. The Enriques surfaces that we investigate include very general $n$-nodal Enriques surfaces and very general cuspidal Enriques surfaces. We also describe the action of the automorphism group on the set of smooth rational curves and on the set of elliptic fibrations.
Umemura Quadric Fibrations and Maximal Subgroups of $Cr_n(\mathbb{C})$
本研究聚焦梅村二次纤维化的等变几何及自同构群极大性，构造出$Cr_n(\mathbb{C})$中无限族两两非共轭的极大连通代数子群，为该领域提供新的子群结构实例。
The Getzler's formula relates the S_n-equivariant Hodge-Deligne polynomial of the space of ordered tuples of distinct points on a given variety X with the Hodge-Deligne polynomial of X. We obtain the analogue of this formula for the case when X has a nontrivial automorphism group. Collecting together all strata of $\mathcal{M}_2$ with different automorphism groups, we derive a formula for the S_n-equivariant Euler characteristic of $\mathcal{M}_{2,n}$.
We give a system of generators of the automorphism group of the singular $K3$ surface of discriminant 7. This system of generators consists of the inversion involutions of some elliptic fibrations with a section together with ${\rm PGL}_2(7)$.
On Néron-Severi lattices of Jacobian elliptic K3 surfaces
本文分类了有限自同构群 K3 曲面及无限自同构群且 Néron-Severi 格为 2-初等情形下具有有限 Mordell-Weil 群的 Jacobian 椭圆纤维化，并计算了相应格论重数。
Consider a fibred compact Kähler manifold X endowed with a relatively ample line bundle, such that each fibre admits a constant scalar curvature Kähler metric and has discrete automorphism group. Assuming the base of the fibration admits a twisted extremal metric where the twisting form is a certain Weil-Petersson type metric, we prove that X admits an extremal metric for polarisations making the fibres small. Thus X admits a constant scalar curvature Kähler metric if and only if the Futaki invariant vanishes. This extends a result of Fine, who proved this result when the base admits no contin…
We describe all the elliptic fibrations with section on the Kummer surface X of the Jacobian of a very general curve C of genus 2 over an algebraically closed field of characteristic 0, modulo the automorphism group of X and the symmetric group on the Weierstrass points of C. In particular, we compute elliptic parameters and Weierstrass equations for the 25 different fibrations and analyze the reducible fibers and Mordell-Weil lattices. This answers completely a question posed by Kuwata and Shioda in 2008.
We show that the automorphism group of a certain subclass of smooth Gizatullin surfaces with a distinguished and rigid extended divisor is generated by automorphisms of A1-fibrations. Moreover, such surfaces provide examples of smooth Gizatullin surfaces with a non-transitive action of the automorphism group. Thus, they represent counterexamples to Gizatullin's conjecture. For such surfaces we give explicit orbits of the natural action of the automorphism group in some special cases. Further, we present their automorphism groups as amalgamated products of two subgroups.
Elliptic Fibrations on Vinberg's Most Algebraic K3 Surface
本文介绍如何利用计算机代数系统 OSCAR 寻找给定曲面上的所有椭圆纤维化并计算其 Weierstrass 模型，并以 Vinberg 最代数 K3 曲面为例进行了演示。
This is mainly a review of my results related to the title. We discuss, how many elliptic fibrations and elliptic fibrations with infinite automorphism group (or the Mordell-Weil group) an algebraic K3 surface over an algebraically closed field can have. This was the subject of my talk at Oberwolfach Workshop "Higher dimensional elliptic fibrations" in October 2010.
In 1980, Gizatullin classified rational surfaces endowed with an automorphism whose action on the Neron-Severi group is parabolic: these surfaces are endowed with an elliptic fibration invariant by the automorphism. The aim of this expository paper is to present for non-experts the details of Gizatullin's original proof, and to provide an introduction to a recent paper by Cantat and Dolgachev.
We classify possible finite groups of symplectic automorphisms of K3 surfaces of order divisible by 11. The characteristic of the ground field must be equal to 11. The complete list of such groups consists of five groups: the cyclic group of order 11, $11\rtimes 5$, $L_2(11)$ and the Mathieu groups $M_{11}$, $M_{22}$. We also show that a surface $X$ admitting an automorphism $g$ of order 11 admits a $g$-invariant elliptic fibration with the Jacobian fibration isomorphic to one of explicitly given elliptic K3 surfaces.
We study the connected algebraic groups acting on Mori fibrations $X \to Y$ with $X$ a rational threefold and $\mathrm{dim}(Y) \geq 1$. More precisely, for these fibre spaces we consider the neutral component of their automorphism groups and study their equivariant birational geometry. This is done using, inter alia, minimal model program and Sarkisov program and allows us to determine the maximal connected algebraic subgroups of $\mathrm{Bir}(\mathbb{P}^3)$, recovering most of the classification results of Hiroshi Umemura in the complex case.
An explicit Enriques surface with an automorphism of minimum entropy
本文借助计算机和椭圆纤维化方法，推导出复 Enriques 曲面上具有最小拓扑熵的 Oguiso-Yu 自同构的显式方程。
We consider two linear reductive algebraic groups $ G $ and $ G' $ over $ C $. Take a finite dimensional rational representation $ W $ of $ G \times G' $. Let $ Y = W // G := Spec C[W]^G $ and $ X = W // G' := \Spec C[W]^{G'} $ be the affine quotients. The quotient space $ X $ (respectively $ Y $) naturally inherits the action of $ G $ (respectively $ G' $). In this note, we study the interrelation between the orbit structures of $ X / G $ and $ Y / G' $. In a good situation, we can embed $ Y / G' $ into $ X / G $, and the embedding map preserves important properties such as the closure relati…
We consider fibrations by affine lines on smooth affine surfaces obtained as complements of smooth rational curves $B$ in smooth projective surfaces $X$ defined over an algebraically closed field of characteristic zero. We observe that except for two exceptions, these surfaces $X \setminus B$ admit infinitely many families of $\mathbb{A}^1$-fibrations over the projective line with irreducible fibers and a unique singular fiber of arbitrarily large multiplicity. For $\mathbb{A}^1$-fibrations over the affine line, we give a new and essentially self-contained proof that the set of equivalence cla…
Equivariant automorphisms of the Cox construction and applications
本文刻画单扇Cox构造关于大环面中足够大的连通复李子群的等变全纯自同构群，并用于确定有理矩角流形及完备单扇环面簇的相应自同构群。
We classify, up to automorphisms, the elliptic fibrations on the singular K3 surface $X$ whose transcendental lattice is isometric to $\langle 6\rangle\oplus \langle 2\rangle$.
We show that automorphism groups of Hopf and Kodaira surfaces have unbounded finite subgroups. For elliptic fibrations on Hopf, Kodaira, bielliptic, and K3 surfaces, we make some observations on finite groups acting along the fibers and on the base of such a fibration.
The aim of this paper is to study algebraic K3 surfaces (defined over the complex number field) with a symplectic automorphism of prime order. In particular we consider the action of the automorphism on the second cohomology with integer coefficients. We determine the invariant sublattice and its perpendicular complement, and show that the latter coincides with the Coxeter-Todd lattice in the case of automorphism of order three. We also compute many explicit examples, with particular attention to elliptic fibrations.
For a variety $X$, a big $\mathbb{Q}$-divisor $L$ and a closed connected subgroup $G \subset \mathrm{Aut}(X, L)$ we define a $G$-invariant version of the $δ$-threshold. We prove that for a Fano variety $(X, -K_X)$ and a connected subgroup $G \subset \mathrm{Aut}(X)$ this invariant characterizes $G$-equivariant uniform $K$-stability. We also use this invariant to investigate $G$-equivariant $K$-stability of some Fano varieties with large groups of symmetries, including spherical Fano varieties. We also consider the case of $G$ being a finite group.
Parabolic automorphisms of hyperkahler manifolds
本文研究超 Kähler 流形上的抛物自同构，证明保持 Lagrangian 纤维化者在纤维上遍历作用，并将 K3 曲面的相关结果推广至一般超 Kähler 情形。
We characterize the irreducible polynomials that occur as a characteristic polynomial of an automorphism of an even unimodular lattice of given signature, generalizing a theorem of Gross and McMullen. As part of the proof, we give a general criterion in terms of Witt groups for a bilinear form equipped with an action of a group G over a discretely valued field to contain a unimodular G-stable lattice.
Let $G$ be a finite group acting effectively on the complex affine plane. If the $G$-action commutes with an étale endomorphism $f$ of the affine plane and the order of $G$ is even then the endomorphism $f$ is an automorphism.
Equivariant CM minimization for extremal manifolds
本文提出在极值凯勒流形中证明等变CM最小化猜想，通过扩展Székelyhidi的结果，允许自同构和极值度量，推广了Hattori对cscK流形的研究。
A Gizatullin surface is a normal affine surface $V$ over $\bf C$, which can be completed by a zigzag; that is, by a linear chain of smooth rational curves. In this paper we deal with the question of uniqueness of $\bf C^*$-actions and $\bf A^1$-fibrations on such a surface $V$ up to automorphisms. The latter fibrations are in one to one correspondence with $\bf C_+$-actions on $V$ considered up to a "speed change". Non-Gizatullin surfaces are known to admit at most one $\bf A^1$-fibration $V\to S$ up to an isomorphism of the base $S$. Moreover an effective $\bf C^{*}$-action on them, if it doe…
Automorphism groups of toroidal horospherical varieties
本文研究了toroidal horospherical varieties的自同构群结构，通过分析Demazure根的特征，给出了判断此类品种自同构群可约性的标准，并应用该结果证明了某些P^1束的K-不稳定性质。
We study equivariant affine embeddings of homogeneous spaces and their equivariant automorphisms. An example of a quasiaffine, but not affine, homogeneous space with finitely many equivariant automorphisms is presented. We prove the solvability of any connected group of equivariant automorphisms for an affine embedding with a fixed point and finitely many orbits. This is applied to studying the orbital decomposition for algebraic monoids and canonical embeddings of quasiaffine homogeneous spaces, i.e., those affine embeddings associated with the coordinate algebras of homogeneous spaces, provi…
Automorphisms of the Quantum Cohomology of the Springer Resolution and Applications
该研究引入作用于Springer解消等变量子上同调的量子Demazure–Lusztig算子，给出其环面等变量子上同调的生成元关系表示与典型类型显式描述，并通过Toda极限复现了Kim关于完全旗簇的相关结果。
Action of Weyl group on equivariant K-theory of flag varieties
本文研究半单线性群Weyl群对旗簇等变K-理论环的作用，利用分差算子和局部化定理计算反射自同构公式，并在Schubert簇结构层类基下展开，给出有效近似及motivic Chern类基下的讨论。
We show that a compact Kaehler 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.
The relative movable cone conjecture for K-trivial fibrations in varieties with well-clipped movable cones
本文证明了在纤维为商积的K-平凡纤维化中，弱相对Kawamata-Morrison可动锥猜想成立，且在特定条件下得出完整猜想，并推导出有限数量的极小模型。
Let $X/\mathbb{P}^1$ be a Mori fibre space with general fibre of Picard rank at least two. We prove that there is a proper closed subset $S\subsetneq X$, invariant by the connected component of the identity ${\rm Aut}^{\circ}(X)$ of the automorphism group of $X$, which is moreover the orbit of a section $s$ and whose intersection with a fibre is an orbit of the subgroup of ${\rm Aut}^{\circ}(X)$ acting trivially on $\mathbb{P}^1$. Such result is a tool to describe equivariant birational maps from $X/\mathbb{P}^1$ to other Mori fibre spaces and therefore finds its applications in the study of c…
We discuss the birational geometry and the Kodaira dimension of certain varieties previously constructed by Schreieder, proving that in any dimension they admit an elliptic fibration and they are not of general type. The $l$-dimensional variety $Y_n^l$, which is the quotient of the product of a certain curve $C_n$ by itself $l$ times by a group $G\simeq\left(\mathbb{Z}/n\mathbb{Z}\right)^{l-1}$ of automorphisms, was constructed by Schreieder to obtain varieties with prescribed Hodge numbers. If $n=3^c$ Schreieder constructed an explicit smooth birational model of it, and Flapan proved: the Kod…
The integral Chow rings of moduli of Weierstrass fibrations
本文计算了射影直线上极小 Weierstrass 纤维化模空间的整系数 Chow 环，利用商栈表示和等变交叠理论确定了各 Chow 环的完整生成元与关系，并对 $N=1$ 和 $N=2$ 的情况给出了更明确的计算。
Automorphism groups and linearizability of rational Fano conic bundle threefolds
本文研究了Fano锥面三维流形的自同构群与线性化问题，证明了No. 2.18类Fano三维流形对于其自同构群的线性化性质。
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