Symplectic automorphisms on Kummer surfaces

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 H²(X,\mathbbZ){H^2(X,\mathbb{Z})} which is fixed by the isometries induced by G. However for certain groups these discriminants are not the same as those found for explicit examples. Here we describe Kummer surfaces for which this phenomena happens and we explain the difference.     

Elliptic K3 Surfaces with Abelian and Dihedral Groups of Symplectic Automorphisms

We analyze K3 surfaces admitting an elliptic fibration ? and a finite group G of symplectic automorphisms preserving this elliptic fibration. We construct the quotient elliptic fibration ?/G comparing its properties to the ones of ?.

We show that if ? 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.     

Equivariant autoequivalences for finite group actions

The familiar Fourier-Mukai technique can be extended to an equivariant setting where a finite group G acts on a smooth projective variety X. In this paper we compare the group of invariant autoequivalences Aut(DbG(X)) with the group of autoequivalences of DG(X). We apply this method in three cases: Hilbert schemes on K3 surfaces, Kummer surfaces and canonical quotients.     

Elliptic Fibrations and Symplectic Automorphisms on K3 Surfaces

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 ⊕ 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.     

Ordinary reduction of K3 surfaces

Let X be a K3 surface over a number field K. We prove that there exists a finite algebraic field extension E/K such that X has ordinary reduction at every non-archimedean place of E outside a density zero set of places.      

On products of graphs and regular groups

A graphX is called a graphical regular representation (GRR) of a groupG if the automorphism group ofX is regular and isomorphic toG. Watkins and Nowitz have shown that the direct productG×H of two finite groupsG andH has aGRR if both factors have aGRR and if at least one factor is different from the cyclic group of order two. We give a new proof of this result, thereby removing the restriction to finite groups. We further show that the complementX′ of a finite or infinite graphX is prime with respect to cartesian multiplication ifX is composite and not one of six exceptional graphs.     

Characters of the infinite symplectic group—A Riesz ring approach

We classify both the finite and infinite characters of the inductive limit symplectic group G. An important feature of our technique is the systematic use of a multiplicative structure on an “ordered completion” of the K₀-group for the group C^*-algebra A of G. We also give explicit examples of the K-theory for certain primitive quotients of A.     

The influence of <Emphasis Type="Italic">X</Emphasis>-semipermutability of subgroups on the structure of finite groups

Let A be a subgroup of a group G and X be a nonempty subset of G. A is said to be X-semipermutable in G if A has a supplement T in G such that A is X-permutable with every subgroup of T. In this paper, we investigate further the influence of X-semipermutability of some subgroups on the structure of finite groups. Some new criteria for a group G to be supersoluble or p-nilpotent are obtained. This work was supported by National Natural Science Foundation of China (Grant Nos. 10771172, 10771180)     

On the number of anisotropic simple submodules in modules with a form

Let G be a finite group, and V a finite-dimensional semisimple G-module over a finite field. Assume that V is endowed with a nonsingular bilinear form which is symmetric or symplectic, and which is invariant under the action of G. In this setting, we compute the number of anisotropic simple submodules of V.Received: 25 May 2004     

The 2-factoriality of the O’Grady moduli spaces

The aim of this work is to show that the moduli space M ₁₀ introduced by O’Grady is a 2-factorial variety. Namely, M ₁₀ is the moduli space of semistable sheaves with Mukai vector v: = (2, 0, −2) in H^ev(X,\mathbbZ){H^{ev}(X,\mathbb{Z})} on a projective K3 surface X. As a corollary to our construction, we show that the Donaldson morphism gives a Hodge isometry between v^{^}{v^{\perp}} (sublattice of the Mukai lattice of X) and its image in H² ([(M)\tilde]₁₀, \mathbbZ){H^{2} (\widetilde{M}_{10}, \mathbb{Z})}, lattice with respect to the Beauville form of the 10-dimensional irreducible symplectic manifold [(M)\tilde]₁₀{\widetilde{M}_{10}}, obtained as symplectic resolution of M ₁₀. Similar results are shown for the moduli space M ₆ introduced by O’Grady to produce its 6-dimensional example of irreducible symplectic variety.     

G‐invariant persistent homology

Classical persistent homology is a powerful mathematical tool for shape comparison. Unfortunately, it is not tailored to study the action of transformation groups that are different from the group Homeo(X) of all self‐homeomorphisms of a topological space X. This fact restricts its use in applications. In order to obtain better lower bounds for the natural pseudo‐distance d_G associated with a group G ? Homeo(X), we need to adapt persistent homology and consider G‐invariant persistent homology. Roughly speaking, the main idea consists in defining persistent homology by means of a set of chains that is invariant under the action of G. In this paper, we formalize this idea and prove the stability of the persistent Betti number functions in G‐invariant persistent homology with respect to the natural pseudo‐distance d_G. We also show how G‐invariant persistent homology could be used in applications concerning shape comparison, when the invariance group is a proper subgroup of the group of all self‐homeomorphisms of a topological space. In this paper, we will assume that the space X is triangulable, in order to guarantee that the persistent Betti number functions are finite without using any tameness assumption. Copyright © 2014 John Wiley & Sons, Ltd.     

A rigidity theorem for automorphism groups of trees

A group G is called unsplittable if Hom(G, ℤ) = 0 and this group is not a non-trivial amalgam. Let X be a tree with a countable number of edges incident at each vertex and G be its automorphism group. In this paper we prove that the vertex stabilizers are unsplittable groups. Bass and Lubotzky proved (see [3]) that for certain locally finite trees X, the automorphism group determines the tree X (that is, knowing the automorphism group we can “construct” the tree X). We generalize this Theorem of Bass and Lubotzky, using the above result. In particular we show that the Theorem holds even for trees which are not locally finite. Moreover, we prove that the permutation group of an infinite countable set is unsplittable and the infinite (or finite) cartesian product of unsplittable groups is an unsplittable group as well. This research was supported by the European Social Fund and National resources-EPEAEK II grant Pythagoras 70/3/7298.     

Affine embeddings with a finite number of orbits

LetG be a reductive algebraic group and letH be a reductive subgroup ofG. We describe all pairs (G, H) such that, for any affineG-varietyX with a denseG-orbit isomorphic toG/H, the number ofG-orbits inX is finite.Work of both authors was supported by INTAS-OPEN-97-1570, by CRDF grant RM1-2088, and by RFBR grant 01-01-00756.     

On a spectral sequence for equivariant K-theory

We apply the “homotopy coniveau” machinery developed by the first-named author to the K-theory of coherent G-sheaves on a finite type G-scheme X over a field, where G is a finite group. This leads to a definition of G-equivariant higher Chow groups (different from the Chow groups of classifying spaces constructed by Totaro and generalized to arbitrary X by Edidin–Graham) and an Atiyah–Hirzebruch spectral sequence from the G-equivariant higher Chow groups to the higher K-theory of coherent G-sheaves on X. This spectral sequence generalizes the spectral sequence from motivic cohomology to K-theory constructed by Bloch–Lichtenbaum and Friedlander–Suslin. The first-named author gratefully acknowledges the support of the Humboldt Foundation through the Wolfgang Paul Program, and support of the NSF via grants DMS-0140445 and DMS-0457195.     

Algebraic Hamiltonian actions

In this paper we deal with a Hamiltonian action of a reductive algebraic group G on an irreducible normal affine Poisson variety X. We study the quotient morphism \({\mu_{G,X}//G : X//G \rightarrow \mathfrak{g} //G}\) of the moment map \({\mu_{G,X} : X\rightarrow \mathfrak{g}}\) . We prove that for a wide class of Hamiltonian actions (including, for example, actions on generically symplectic varieties) all fibers of the morphism μ _G,X //G have the same dimension. We also study the “Stein factorization” of μ _G,X //G. Namely, let C _G,X denote the spectrum of the integral closure of \({\mu_{G,X}^{*}(\mathbb{K}[\mathfrak{g}]^G)}\) in \({\mathbb{K}(X)^G}\) . We investigate the structure of the \({\mathfrak{g} //G}\) -scheme C _G,X . Our results partially generalize those obtained by F. Knop for the actions on cotangent bundles and symplectic vector spaces.     

Symplectic involutions on deformations of K3[2]

Let X be a hyperk?hler manifold deformation equivalent to the Hilbert square of a K3 surface and let φ be an involution preserving the symplectic form. We prove that the fixed locus of φ consists of 28 isolated points and one K3 surface, and moreover that the anti-invariant lattice of the induced involution on H ₂(X, ℤ) is isomorphic to E ₈(−2). Finally we show that any couple consisting of one such manifold and a symplectic involution on it can be deformed into a couple consisting of the Hilbert square of a K3 surface and the involution induced by a symplectic involution on the K3 surface.     

Stopped Markov chains with stationary occupation times

Summary. Let E be a finite set equipped with a group G of bijective transformations and suppose that X is an irreducible Markov chain on E that is equivariant under the action of G. In particular, if E=G with the corresponding transformations being left or right multiplication, then X is a random walk on G. We show that when X is started at a fixed point there is a stopping time U such that the distribution of the random vector of pre-U occupation times is invariant under the action of G. When G acts transitively (that is, E is a homogeneous space), any non-zero, finite expectation stopping time with this property can occur no earlier than the time S of the first return to the starting point after all states have been visited. We obtain an expression for the joint Laplace transform of the pre-S occupation times for an arbitrary finite chain and show that even for random walk on the group of integers mod r the pre-S occupation times do not generally have a group invariant distribution. This appears to contrast with the Brownian analog, as there is considerable support for the conjecture that the field of local times for Brownian motion on the circle prior to the counterpart of S is stationary under circular shifts. Received: 6 December 1995 / In revised form: 11 June 1997     

A Boolean algebra of characteristic subgroups of a finite group

We construct a “natural” sublattice L(G) of the lattice of all of those subgroups of a finite group G that contain the Frattini subgroup F(G){\Phi(G)} . We show that L(G) is a Boolean algebra, and that its members are characteristic subgroups of G. If F(G){\Phi(G)} is trivial, then L(G) is exactly the set of direct factors U of G such that U and G/U have no common nontrivial homomorphic image.     

On Injective and Cofibrant Equivariant Minimal Models

Let X be a G-connected nilpotent simplicial set, where G is a finite Hamiltonian group. We construct a cofibrant equivariant minimal model of X with the strong homotopy type of the injective minimal model of X defined by Triantafillou.