Group Of Symplectic Automorphisms Research Articles

Smooth constructions of homotopy-coherent actions

We prove that, for nice classes of infinite-dimensional smooth groups G, natural constructions in smooth topology and symplectic topology yield homotopically coherent group actions of G. This yields a bridge between infinite-dimensional smooth groups and homotopy theory. The result relies on two computations: One showing that the diffeological homotopy groups of the Milnor classifying space BG are naturally equivalent to the (continuous) homotopy groups, and a second showing that a particular strict category localizes to yield the homotopy type of BG. We then prove a result in symplectic geometry: These methods are applicable to the group of Liouville automorphisms of a Liouville sector. The present work is written with an eye toward [OT19], where our constructions show that higher homotopy groups of symplectic automorphism groups map to Fukaya-categorical invariants, and where we prove a conjecture of Teleman from the 2014 ICM in the Liouville and monotone settings.

Conway subgroup symmetric compactifications redux

We extend the investigation in [1] of special toroidal compactifications of heterotic string theory for which the half-BPS states provide representations of subgroups of the Conway group. We also explore dual descriptions of these theories and find that they are all linked to either F-theory or type IIA string theory on K3 surfaces with symplectic automorphism groups that are the same Conway subgroups as those of the heterotic dual. The matching with type IIA K3 dual theories includes both the matching of symmetry groups and a comparison between the Narain lattice on the heterotic side and the cohomology lattice on the type IIA side. We present twelve examples where we can identify a type IIA dual K3 orbifold theory as the dual description of the heterotic theory. In addition, we include a supplementary Mathematica package that performs most of the computations required for these comparisons.

Automorphisms and periods of cubic fourfolds

We classify the symplectic automorphism groups for cubic fourfolds. The main inputs are the global Torelli theorem for cubic fourfolds and the classification of the fixed-point sublattices of the Leech lattice. Among the highlights of our results, we note that there are 34 possible groups of symplectic automorphisms, with 6 maximal cases. The six maximal cases correspond to 8 non-isomorphic, and isolated in moduli, cubic fourfolds; six of them previously identified by other authors. Finally, the Fermat cubic fourfold has the largest possible order (174, 960) for the automorphism group (non-necessarily symplectic) among all smooth cubic fourfolds.

Classification of degenerations and Picard lattices of Kählerian K3 surfaces with symplectic automorphism group

In [1]–[6] we classified the degenerations and Picard lattices of Kählerian K3 surfaces with finite symplectic automorphism groups of high order. This classification was not considered for the remaining groups of small order (, , , , and ) because each of these cases requires very long and difficult considerations and calculations. Here we consider this classification for the dihedral group of order .

Classification of Degenerations and Picard Lattices of Kählerian K3 Surfaces with Symplectic Automorphism Group C4

In the author’s papers of 2013–2018, the degenerations and Picard lattices of Kählerian K3 surfaces with finite symplectic automorphism groups of high order were classified. For the remaining groups of small order—D6, C4, (C2)2, C3, C2, and C1—the classification was not completed because each of these cases requires very long and difficult considerations and calculations. The case of D6 was recently completely studied in the author’s paper of 2019. In the present paper an analogous complete classification is presented for the cyclic group C4 of order 4.

Finite groups of symplectic automorphisms of hyperkahler manifolds of type K3

We determine the possible finite groups $G$ of symplectic automorphisms of hyperkahler manifolds which are deformation equivalent to the second Hilbert scheme of a K3 surface. We prove that $G$ has such an action if, and only if, it is isomorphic to a subgroup of either the Mathieu group $M_{23}$ having at least four orbits in its natural permutation representation on $24$ elements, or one of two groups $3^{1+4}{:}2.2^2$ and $3^4{:}A_6$ associated to $\mathcal{S}$-lattices in the Leech lattice. We describe in detail those $G$ which are maximal with respect to these properties, and (in most cases) we determine all deformation equivalence classes of such group actions. We also compare our results with the predictions of Mathieu Moonshine.

On generalized Kummer surfaces and the orbifold Bogomolov-Miyaoka-Yau inequality

A generalized Kummer surface X = Km(T, G) is the resolution of a quotient of a torus T by a finite group of symplectic automorphisms G. We complete the classification of generalized Kummer surfaces by studying the two last groups which have not been yet studied. For these surfaces we compute the associated Kummer lattice K G , which is the minimal primitive sub-lattice containing the exceptional curves of the resolution X → T /G. We then prove that a K3 surface is a generalized Kummer surface of type Km(T, G) if and only if its Neron-Severi group contains K G. For smooth-orbifold surfaces X of Kodaira dimension ≥ 0, Kobayashi proved the orbifold Bogomolov Miyaoka Yau inequality c 2 1 (X) ≤ 3c 2 (X). For Kodaira dimension 2, the case of equality is characterized as X being uniformized by the complex 2-ball B 2. For smooth-orbifold K3 and Enriques surfaces we characterize the case of equality as being uniformized by C 2 .

Classification of Picard lattices of K3 surfaces

Using the results of our papers [1]–[4] on the classification of degenerations of Kählerian K3 surfaces, we classify the Picard lattices of Kählerian K3 surfaces. By classification we mean classification depending on their possible finite symplectic automorphism groups and their non- singular rational curves when the Picard lattice is negative definite.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. III

Following our papers [1] and [2] (Parts I and II), we classify degenerations of codimension 2 or more of Kählerian surfaces with finite symplectic automorphism groups. In [1] and [2] this was done for codimension .

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. II

We prove the main conjecture (Conjecture 7.1) of the paper [1]. Using this result, we classify degenerations of codimension 1 of Kählerian K3 surfaces with finite symplectic automorphism groups.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups. II

Доказывается основная гипотеза 7.1 статьи [1]. С помощью этого результата классифицируются вырождения коразмерности 1 кэлеровых K3-поверхностей с конечными симплектическими группами автоморфизмов. Библиография: 20 наименований.

Towards a classification of symplectic automorphisms on manifolds of $$K3^{[n]}$$ K 3 [ n ] type

The present paper is devoted to the classification of symplectic automorphisms of some hyperkahlermanifolds. The result presented here is a proof that all finite groups of symplectic automorphisms of manifolds of \(K3^{[n]}\) typeare contained in Conway’s group \(Co_1\).

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups

Using the results of [1]–[3] on Kählerian K3 surfaces and Niemeier lattices, we classify degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups with emphasis on degenerations of codimension 1.

Degenerations of Kählerian K3 surfaces with finite symplectic automorphism groups

При использовании результатов из [1]-[3], где изучались кэлеровы К3-поверхности и решетки Нимейера, классифицируются вырождения кэлеровых К3-поверхностей с конечными симплектическими группами автоморфизмов. Главным образом, классифицируются вырождения коразмерности 1. Библиография: 23 наименования.

Exotic iterated Dehn twists

Consider cotangent bundles of exotic spheres, with their canonical symplectic structure. They admit automorphisms which preserve the part at infinity of one fibre, and which are analogous to the square of a Dehn twist. Pursuing that analogy, we show that they have infinite order up to isotopy (inside the group of all automorphisms with the same behaviour).

The overarching finite symmetry group of Kummer surfaces in the Mathieu group M 24

In view of a potential interpretation of the role of the Mathieu group M_24 in the context of strings compactified on K3 surfaces, we develop techniques to combine groups of symmetries from different K3 surfaces to larger 'overarching' symmetry groups. We construct a bijection between the full integral homology lattice of K3 and the Niemeier lattice of type (A_1)^24, which is simultaneously compatible with the finite symplectic automorphism groups of all Kummer surfaces lying on an appropriate path in moduli space connecting the square and the tetrahedral Kummer surfaces. The Niemeier lattice serves to express all these symplectic automorphisms as elements of the Mathieu group M_24, generating the 'overarching finite symmetry group' (Z_2)^4:A_7 of Kummer surfaces. This group has order 40320, thus surpassing the size of the largest finite symplectic automorphism group of a K3 surface by orders of magnitude. For every Kummer surface this group contains the group of symplectic automorphisms leaving the Kaehler class invariant which is induced from the underlying torus. Our results are in line with the existence proofs of Mukai and Kondo, that finite groups of symplectic automorphisms of K3 are subgroups of one of eleven subgroups of M_23, and we extend their techniques of lattice embeddings for all Kummer surfaces with Kaehler class induced from the underlying torus.

Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

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, and we describe the coarse moduli space of K3 surfaces with a given finite group contained in the Mordell–Weil group. Considering automorphisms coming from the base of the fibration, we find the Mordell–Weil lattice of a fibration described by Kloosterman, and we find K3 surfaces with dihedral groups as group of symplectic automorphisms. We prove the isometries between lattices described by the author and Sarti and lattices described by Shioda and by Greiss and Lam.

The dihedral group $\mathcal D_{5}$ as a group of symplectic automorphisms on K3 surfaces

We prove that if a K3 surface X admits Z/5Z as a group of symplectic automorphisms, then it actually admits D 5 as a group of symplectic automorphisms. The orthogonal complement to the D 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.

K3 surfaces with a symplectic automorphism of order 11

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 C11 of order 11, C11 ⋊ C5 , PSL2(F11) 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.

Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic

We show that Mukai's classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic p under assumptions that (i) the order of the group is coprime to p and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without assumption (ii) we classify all possible new groups which may appear. We prove that assumption (i) on the order of the group is always satisfied if p > 11. For p = 2,3,5,11, we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by p which are not contained in Mukai's list.