We prove that if a K3 surface $X$ admits $\Z/5\Z$ as group of symplectic automorphisms, then it actually admits $\Dh_5$ as group of symplectic automorphisms. The orthogonal complement to the $\Dh_5$-invariants in the second cohomology group of $X$ is a rank 16 lattice, $L$. It is known that $L$ does not depend on $X$: we prove that it is isometric to a lattice recently described by R. L. Griess Jr. and C. H. Lam. We also give an elementary construction of $L$.
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…
It is known that an automorphism group of a K3 surface with Picard number two is either infinite cyclic group or infinite dihedral group if it is infinite. In this paper, we study the generators of an automorphism group. We use the eigenvector corresponding to the spectral radius of an automorphism of infinite order to determine the generators. We also determine whether an automorphism group is the infinite cyclic group or the infinite dihedral group with several examples.
In this paper we investigate when the generic member of a family of K3 surfaces admitting a non--symplectic automorphism of finite order admits also a symplectic automorphism of the same order. We give a complete answer to this question if the order of the automorphism is a prime number and we provide several examples and partial results otherwise. Moreover we prove that, under certain conditions, a K3 surface admitting a non--symplectic automorphism of prime odd order, $p$, also admits a non--symplectic automorphism of order $2p$. This generalizes a previous result by J. Dillies for $p=3$.
We classify the polarized symplectic automorphisms of Fano varieties of smooth cubic fourfolds (equipped with the Plücker polarization) and study the fixed loci.
The present paper is devoted to the classification of symplectic automorphisms of some hyperkähler manifolds. The results contained here are an explicit classification of prime order automorphisms on manifolds of $K3^{[n]}$ type and a proof that all finite groups of symplectic automorphisms of such manifolds are contained in Conway's group $Co_1$.
We treat non-symplectic automorphisms on $K3$ surfaces which act trivially on the Néron-Severi lattice. In this paper, we classify non-symplectic automorphisms of prime-power order, especially 2-power order on $K3$ surfaces, i.e., we describe their fixed locus.
In this note, we treat a pair of a K3 surface and a non-symplectic automorphism of order 7m (m=1, 3 and 6) on it. We show that if the fixed locus of a non-symplectic automorphism order 7 is "special" then the pair is unique up to isomorphism. And we describe fixed loci of non-symplectic automorphisms of order 21 and 42.
Let X be an irreducible smooth complex projective curve of genus at least 3. Fix a line bundle L on X. Let M_{Sp}(L) be the moduli space of symplectic bundles (E, ExE ---> L) on X, with the symplectic form taking values in L. We show that the automorphism group of M_{Sp}(L) is generated by automorphisms sending E to ExM, where M is a 2-torsion line bundle, and automorphisms induced by automorphisms of X.
We survey recent results on ample cones and birational contractions of holomorphic symplectic varieties of K3 type, focusing on explicit constructions and concrete examples.
We classify primitive non-symplectic automorphisms of order 6 on K3 surfaces. We show how their study can be reduced to the study of non-symplectic automorphisms of order 3 and to a local analysis of the fixed loci. In particular, we determine the possible fixed loci and show that when the Picard lattice is fixed, K3 surfaces come in mirror pairs.
An automorphism of order $n$ of a K3 surface is called purely non-symplectic if it multiplies the holomorphic symplectic form by a primitive $n$-th root of unity. We give the classification of purely non-symplectic automorphisms with $φ(n)\geq 12$ where $φ$ denotes the Euler totient function.
The present paper proves that finite symplectic groups of automorphisms of hyperkähler fourfolds deformation equivalent to the Hilbert scheme of two points on a $K3$ surface are contained in the simple group $Co_1$. Then we give an example of a symplectic automorphism of order 11 on the Fano scheme of lines of a cubic fourfold.
In this paper we study K3 surfaces with a non-symplectic automorphism of order 3. In particular, we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology. This allows us to describe the structure of the moduli space and to show that it has three irreducible components.
It is observed that the recent result of Voisin and earlier ones of the author suffice to prove in complete generality that symplectic automorphisms of finite order of a K3 surface X act as identity on the Chow group CH^2(X) of zero-cycles.
In this paper, we study non-symplectic automorphisms of order 3 on algebraic $K3$ surface over $\mathbb{C}$ which act trivially on the Néron-Severi lattice. In particular we shall characterize their fixed locus in terms of the invariants of 3-elementary lattices.
In the present paper we prove that finite symplectic groups of automorphisms of manifolds of k3^[n] type can be obtained by deforming natural morphisms arising from K3 surfaces if and only if they satisfy a certain numerical condition.
We classify non-symplectic automorphisms of odd prime order on irreducible holomorphic symplectic manifolds which are deformations of Hilbert schemes of any number n of points on K3 surfaces, extending results already known for n=2. In order to do so, we study the properties of the invariant lattice of the automorphism (and its orthogonal complement) inside the second cohomology lattice of the manifold. We also explain how to construct automorphisms with fixed action on cohomology: in the cases n=3,4 the examples provided allow to realize all admissible actions in our classification. For n=4, …
We prove that for any polarized symplectic automorphism of the Fano variety of lines of a cubic fourfold (equipped with the Plücker polarization), the induced action on the Chow group of 0-cycles is identity, as predicted by Bloch-Beilinson conjecture.
Let $X$ be a hyperkähler variety, and assume that $X$ admits a non-symplectic automorphism $σ$ of order $k>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<σ>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain cubic fourfolds having an order $3$ non-symplectic automorphism.
We prove the main Conjecture 4 of our paper arXiv:1403.6061v5, which leads to classification of degenerations of codimension one of Kahlerian K3 surfaces with finite symplectic automorphism groups.
We give a necessary and sufficient condition for an automorphism of the Hilbert scheme of points on a K3 surface (non necessarily algebraic) to be induced by an automorphism of the surface. We prove furthermore that the group of birational transformations of a projective irreducible holomorphic symplectic manifold is finitely generated.
Let $X$ be a hyperkähler variety, and assume $X$ has a non-symplectic automorphism $σ$ of order $>{1\over 2}\dim X$. Bloch's conjecture predicts that the quotient $X/<σ>$ should have trivial Chow group of $0$-cycles. We verify this for Fano varieties of lines on certain special cubic fourfolds having an order $3$ non--symplectic automorphism.
In this paper, we prove that, over an algebraically closed field whose characteristic is not 2,3 nor 7, a pair of a K3 surface and a purely non-symplectic automorphism of order 21 or 42 is unique up to isomorphism.
We classify non symplectic prime order automorphisms and all finite order symplectic automorphism groups of generalised Kummer fourfolds using lattice theory and recent results on ample cones and monodromy groups. We study various geometric realisations of the automorphisms obtained in the classification. In the case of higher dimensional generalised Kummers, we provide full results for finite symplectic automorphisms groups and we sketch the classification for prime order non-symplectic groups.
In this note we present the classification of non-symplectic automorphisms of prime order on K3 surfaces, i.e.we describe the topological structure of their fixed locus and determine the invariant lattice in cohomology. We provide new results for automorphisms of order 5 and 7 and alternative proofs for higher orders. Moreover, for any prime p, we identify the irreducible components of the moduli space of K3 surfaces with a non-symplectic automorphism of order p.
In this paper we present a classification of non-symplectic automorphisms of K3 surfaces whose order is a multiple of seven by describing the topological type of their fixed locus. In the case of purely non-symplectic automorphisms, we provide new results for order 14 and alternative proofs for orders 21, 28 and 42, so that we can unify in the same paper the results on these automorphisms. For each of these orders we also consider not purely non-symplectic automorphisms and obtain a complete characterization of their fixed loci. Several results of our paper were obtained independently in a rec…
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.
We show that the Mathieu groups $M_{22}$ and $M_{11}$ can act on the supersingular $K3$ surface with Artin invariant 1 in characteristic 11 as symplectic automorphisms. More generally we show that all maximal subgroups of the Mathieu group $M_{23}$ with three orbits on 24 letters act on a supersingular $K3$ surface with Artin invariant 1 in a suitable characteristic.
We introduce the notion of induced automorphisms in order to state a criterion to determine whether a given automorphism on a manifold of $K3^{[n]}$ type is, in fact, induced by an automorphism of a $K3$ surface and the manifold is a moduli space of stable objects on the $K3$. This criterion is applied to the classification of non-symplectic prime order automorphisms on manifolds of $K3^{[2]}$ type and we prove that almost all cases are covered. Variations of this notion and the above criterion are introduced and discussed for the other known deformation types of irreducible symplectic manifol…
Using results of our preprint "Kahlerian K3 surfaces and Niemeier lattices" arXiv:1109.2879 (and the corresponding papers), we classify degenerations of Kahlerian K3 surfaces with finite symplectic automorphism groups.
We classify K3 surfaces with non-symplectic automorphism of order 16 in full generality. We show that the fixed locus contains only rational curves and points and we completely classify the seven possible configurations. If the Néron-Severi group has rank 6, there are two possibilities and if its rank is 14, there are five possibilities. In particular if the action of the automorphism is trivial on the Néron-Severi group, then we show that its rank is six.
In this paper we classify complex K3 surfaces with non-symplectic automorphism of order 8 that leaves invariant a smooth elliptic curve. We show that the rank of the Picard group is either 10, 14 or 18 and the fixed locus is the disjoint union of elliptic curves, rational curves and points, whose number does not exceed 1, 2, respectively 14. We give examples corresponding to several types of fixed locus in the classification.
Nikulin proved that the isometries induced on the second cohomology group of a K3 surface $X$ by a finite abelian group $G$ of symplectic automorphisms are essentially unique. Moreover he computed the discriminant of the sublattice of $H^2(X, \Z)$ which is fixed by the isometries induced by $G$. However for certain groups these discriminants are not the same of those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.
In this paper we consider the class of K3 surfaces defined as hypersurfaces in weighted projective space, and admitting a non-symplectic automorphism of non-prime order, excluding the orders 4, 8, and 12. We show that on these surfaces the Berglund-Hübsch-Krawitz mirror construction and mirror symmetry for lattice polarized K3 surfaces constructed by Dolgachev agree; that is, both versions of mirror symmetry define the same mirror K3 surface.
The aim of this paper is to generalize results known for the symplectic involutions on K3 surfaces to the order 3 symplectic automorphisms on K3 surfaces. In particular, we will explicitly describe the action induced on the lattice $Λ_{K3}$, isometric to the second cohomology group of a K3 surface, by a symplectic automorphism of order 3; we exhibit the maps $π_*$ and $π^*$ induced in cohomology by the rational quotient map $π:X\dashrightarrow Y$, where $X$ is a K3 surface admitting an order 3 symplectic automorphism $σ$ and $Y$ is the minimal resolution of the quotient $X/σ$; we deduce the re…
Similarly to our papers I and II on the subject (see arXiv:1403.6061 and arXiv:1504.00326), we classify degenerations of codimension 2 and higher of Kahlerian K3 surfaces with finite symplectic automorphism groups. In parts I and II, it was done for the codimension one.
We classify all the K3 surfaces which are minimal models of the quotient of the product of two curves $C_1\times C_2$ by the diagonal action of either the group $\Z/p\Z$ or the group $\Z/2p\Z$. These K3 surfaces admit a non-symplectic automorphism of order $p$ induced by an automorphism of one of the curves $C_1$ or $C_2$. We prove that most of the K3 surfaces admitting a non-symplectic automorphism of order $p$ (and in fact a maximal irreducible component of the moduli space of K3 surfaces with a non-symplectic automorphism of order $p$) are obtained in this way.\\ In addition, we show that o…
We consider the symplectic action of a finite group G on a K3 surface. The Picard group of the K3 surface has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then describe moduli spaces of K3 surfaces with symplectic G-action, extending results of Nikulin in the abelian case. We use our moduli spaces to develop techniques for classifying all possible symplectic actions of a group G.
Let $K$ be any field and $G$ be a finite group. Let $G$ act on the rational function field $K(x_g: \ g \in G)$ by $K$-automorphisms defined by $g \cdot x_h=x_{gh}$ for any $g, \ h \in G$. Denote by $K(G)$ the fixed field $K(x_g: \ g \in G)^G$. Noether's problem asks whether $K(G)$ is rational (=purely transcendental) over $K$. We will give a brief survey of Noether's problem for abelian groups and dihedral groups, and will show that $\Bbb Q(D_n)$ is rational over $\Bbb Q$ for $n \le 10$.
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.
The moduli space of K3 surfaces $X$ with a purely non-symplectic automorphism $σ$ of order $n\geq 2$ is one dimensional exactly when $φ(n)=8$ or $10$. In this paper we classify and give explicit equations for the very general members $(X,σ)$ of the irreducible components of maximal dimension of such moduli spaces. In particular we show that there is a unique one-dimensional component for $n=20,22, 24$, three irreducible components for $n=15$ and two components in the remaining cases.
Let X be an irreducible symplectic manifold and L a nef line bundle on X which is isotropic with respect to the Beauville-Bogomolov quadratic form. It is known that a subgroup Aut(X,L) of an automorphism group of X which fix L is almost abelian. We give a formula of the rank of Aut(X,L) in terms of MBM divisors. We also prove that the nef cone of X cut out MBM classes, which is a generalization of Kovac's structure theorem of nef cones of K3 surfaces
Let $X$ be a hyperkähler variety, and let $G$ be a group of finite order non-symplectic automorphisms of $X$. Beauville's conjectural splitting property predicts that each Chow group of $X$ should split in a finite number of pieces. The Bloch-Beilinson conjectures predict how $G$ should act on these pieces of the Chow groups: certain pieces should be invariant under $G$, while certain other pieces should not contain any non-trivial $G$-invariant cycle. We can prove this for two pieces of the Chow groups when $X$ is the Hilbert scheme of a $K3$ surface and $G$ consists of natural automorphisms.…
The present paper provides several results on automorphisms of hyperkähler (or irreducible holomorphic symplectic) manifolds. In particular it focuses on the symplectic case and contains a classification of prime order symplectic automorphisms on deformations of the Hilbert scheme of two points on a K3 surface and, with additional hypothesis, also on Hilbert schemes of n points on a K3 surface. We also give numerical conditions to establish whether a symplectic automorfism of order 2,3 and 5 on a deformation of the Hilbert scheme of two points on a K3 surface is standard. Moreover we also give…
In this note we interpret a recent result of Gaberdiel, Hohenegger and Volpato in terms of derived equivalences of K3 surfaces. We prove that there is a natural bijection between subgroups of the Conway group Co_1 with invariant lattice of rank at least four and groups of symplectic derived equivalences of D(Coh(X)) of projective K3 surfaces fixing a stability condition.
We describe a compactification by KSBA stable pairs of the five-dimensional moduli space of K3 surfaces with purely non-symplectic automorphism of order four and $U(2)\oplus D_4^{\oplus2}$ lattice polarization. These K3 surfaces can be realized as the minimal resolution of the double cover of $\mathbb{P}^1\times\mathbb{P}^1$ branched along a specific $(4,4)$ curve. We show that, up to a finite group action, this stable pair compactification is isomorphic to Kirwan's partial desingularization of the GIT quotient $(\mathbb{P}^1)^8//\mathrm{SL}_2$ with the symmetric linearization.
In this article, we focus on a new perspective of automorphisms of complex 2-tori, reviewing previous works from a lattice-theoretic point of view. In particular, we give a classification of families of symplectic and non-symplectic automorphisms of prime order.
In this note we discuss techniques for determining the automorphism group of a genus $g$ hyperelliptic curve $\X_g$ defined over an algebraically closed field $k$ of characteristic zero. The first technique uses the classical $GL_2 (k)$-invariants of binary forms. This is a practical method for curves of small genus, but has limitations as the genus increases, due to the fact that such invariants are not known for large genus. The second approach, which uses dihedral invariants of hyperelliptic curves, is a very convenient method and works well in all genera. First we define the normal decompo…
Given $X$ a K3 surface admitting a symplectic automorphism $τ$ of order 4, we describe the isometry $τ^*$ on $H^2(X,\mathbb Z)$. Having called $\tilde Z$ and $\tilde Y$ respectively the minimal resolutions of the quotient surfaces $Z=X/τ^2$ and $Y=X/τ$, we also describe the maps induced in cohomology by the rational quotient maps $X\rightarrow\tilde Z,\ X\rightarrow\tilde Y$ and $\tilde Y\rightarrow\tilde Z$: with this knowledge, we are able to give a lattice-theoretic characterization of $\tilde Z$, and find the relation between the Néron-Severi lattices of $X,\tilde Z$ and $\tilde Y$ in the …
We give a classification of all non-symplectic automorphisms of prime order p acting on irreducible holomorphic symplectic fourfolds deformation equivalent to the Hilbert scheme of two points on a K3 surface, for p=2,3 and 7\leq p \leq 19. Our classification relates the isometry classes of two natural lattices associated to the action of the automorphism on the second cohomology group with integer coefficients with some invariants of the fixed locus and we provide explicit examples. As an application, we find new examples of non-natural non-symplectic automorphisms.
