Kac–Moody $${\widetilde{E}_k}$$-bundles over elliptic curves and del Pezzo surfaces with singularities of type A

Given an elliptic curve Σ, flat E _k -bundles over Σ are in one-to-one correspondence with smooth del Pezzo surfaces of degree 9 − k containing Σ as an anti-canonical curve. This correspondence was generalized to Lie groups of any type. In this article, we show that there is a similar correspondence between del Pezzo surfaces of degree 0 with an A _d -singularity containing Σ as an anti-canonical curve and Kac–Moody [(E)\tilde]_k{\widetilde{E}_{k}}-bundles over Σ with k = 8 − d. In the degenerate case where surfaces are rational elliptic surfaces, the corresponding [(E)\tilde]_k{\widetilde{E}_k}-bundles over Σ can be reduced to E _k -bundles.     

Transcendence degree of zero-cycles and the structure of Chow motives

In the paper we introduce a transcendence degree of a zero-cycle on a smooth projective variety X and relate it to the structure of the motive of X. In particular, we show that in order to prove Bloch’s conjecture for a smooth projective complex surface X of general type with p _g = 0 it suffices to prove that one single point of a transcendence degree 2 in X(ℂ), over the minimal subfield of definition k ⊂ ℂ of X, is rationally equivalent to another single point of a transcendence degree zero over k. This can be of particular interest in the context of Bloch’s conjecture for those surfaces which admit a concrete presentation, such as Mumford’s fake surface, see [Mumford D., An algebraic surface with K ample, (K ²) = 9, p _g = q = 0, Amer. J. Math., 1979, 101(1), 233–244].     

Moment convergence rates in the law of the logarithm for dependent sequences

Let {X _n ; n ≥ 1} be a strictly stationary sequence of negatively associated random variables with mean zero and finite variance. Set S _n = Σ _k=1 ⁿ X _k , M _n = max _k≤n |S _k |, n ≥ 1. Suppose σ ² = EX ₁² + 2Σ _k=2^∞ EX ₁ X _k (0 < σ < ∞). In this paper, the exact convergence rates of a kind of weighted infinite series of E{M _n −σɛ√n log n}₊ and E{|S _n | − σɛ√n log n}₊ as ɛ ↘ 0 and E{σɛ√π ² π/8logn − M _n }₊ as ɛ ↗ ∞ are obtained.     

Discretization of Compact Riemannian Manifolds Applied to the Spectrum of Laplacian

For κ ⩾ 0 and r₀ > 0 let ℳ(n, κ, r₀) be the set of all connected, compact n-dimensional Riemannian manifolds (Mⁿ, g) with Ricci (M, g) ⩾ −(n−1) κ g and Inj (M) ⩾ r₀. We study the relation between the kth eigenvalue λ_k(M) of the Laplacian associated to (Mⁿ,g), Δ = −div(grad), and the kth eigenvalue λ_k(X) of a combinatorial Laplacian associated to a discretization X of M. We show that there exist constants c, C > 0 (depending only on n, κ and r₀) such that for all M ∈ ℳ(n, κ, r₀) and X a discretization of for all k < |X|. Then, we obtain the same kind of result for two compact manifolds M and N ∈ ℳ(n, κ, r₀) such that the Gromov–Hausdorff distance between M and N is smaller than some η > 0. We show that there exist constants c, C > 0 depending on η, n, κ and r₀ such that for all . Mathematics Subject Classification (2000): 58J50, 53C20 Supported by Swiss National Science Foundation, grant No. 20-101 469     

On multivalued topologies on <Emphasis Type="Italic">L</Emphasis>-powersets of multivalued sets

Given an M-valued equality E: X×X → M on a set X, we extend it to the M-valued equality ε: L ^X × L ^X → M on the L-powerset L ^X of X, where L is a complete sublattice of a GL-monoid M. As a result, we come to a category SET(M,L) whose objects are quadruples (X,E,L ^X , ε). This category serves as a ground category for the category L-TOP(M) of (L,M)-valued topological spaces and some of its subcategories, which are the main subject of this paper. In particular, as special cases, we obtain here Chang-Goguen, Lowen, Kubiak-Šostak, and some other known categories related to fuzzy topology. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 4, pp. 237–247, 2005.     

Nonvanishing derivatives and normal families

We consider the differential operators Ψ _k , defined by Ψ₁(y) =y and Ψ _k+1(y)=yΨ _k y+d/dz(Ψ _k (y)) fork ∈ ℕ fork∈ ℕ. We show that ifF is meromorphic in ℂ and Ψ _k F has no zeros for somek≥3, and if the residues at the simple poles ofF are not positive integers, thenF has the formF(z)=((k-1)z+a)/(z ²+β z+γ) orF(z)=1/(az+β) where α, β, γ ∈ ℂ. If the residues at the simple poles ofF are bounded away from zero, then this also holds fork=2. We further show that, under suitable additional conditions, a family of meromorphic functionsF is normal if each Ψ _k (F) has no zeros. These conditions are satisfied, in particular, if there exists δ>0 such that Re (Res(F, a)) <−δ for all polea of eachF in the family. Using the fact that Ψ _k (f ^′/f) =f ^(k)/f, we deduce in particular that iff andf ^(k) have no zeros for allf in some familyF of meromorphic functions, wherek≥2, then {f ^′/f :f ∈F} is normal. The first author is supported by the German-Israeli Foundation for Scientific Research and Development G.I.F., G-643-117.6/1999, and INTAS-99-00089. The second author thanks the DAAD for supporting a visit to Kiel in June–July 2002. Both authors thank Günter Frank for helpful discussions.     

Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

Given a smooth projective toric variety X, we construct an A_∞ category of Lagrangians with boundary on a level set of the Landau–Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X.      

Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra A = L \rtimes GA = \Lambda \rtimes G where L = k[X₁, ..., X_m]/(X₁^l, ..., X_m^l), G = (\mathbbZ/l\mathbbZ)^m\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m and char k does not divide ℓ, in terms of certain subalgebras of A playing the role of “cyclic shifted subgroups”. We show that the rank variety of a finitely generated module M is homeomorphic to the support variety of M defined in terms of the action of the cohomology algebra of A. As an application we derive a theory of rank varieties for the algebra Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.     

On the commutant of certain multiplication operators on spaces of analytic functions

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofM _Z ² (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantM _Z ² ifSM _Z ^2k+1−M _Z ^2k+1 S is compact for some nonnegative integerk, thenS=M _ϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofM _Z ⁿ defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formM _ϕ. Research supported by the Shiraz University Grant 78-SC-1188-657.     

Ordered overlaps in disordered mean-field models

Let M(N) be a sequence of integers with M→∞ as N→∞ and M=o(N). For bounded i.i.d. r.v. ξ _i ^k and bounded i.i.d. r.v. σ _i , we study the large deviation of the family of (ordered) scalar products X ^k =N ⁻¹∑ _{i =1} ^N σ _i ξ _i ^k ,k≤M, under the distribution conditioned on the ξ _i ^k 's. To get a full large deviation principle, it is necessary to specify also the total norm(∑ _{k ≤ M} (X ^k )²)^1/2, which turns to be associated with some extra Gaussian distribution. Our results apply to disordered, mean-field systems, including generalized Hopfield models in the regime of a sublinear number of patterns. We build also a class of examples where this norm is the crucial order parameter. Received: 6 April 1999 / Revised version: 29 May 2000 /?Published online: 24 July 2001     

On the derived category of a regular toric scheme

Let X be a quasi-compact scheme, equipped with an open covering by affine schemes U _σ = Spec A ^σ . A quasi-coherent sheaf on X gives rise, by taking sections over the U _σ , to a diagram of modules over the coordinate rings A ^σ , indexed by the intersection poset Σ of the covering. If X is a regular toric scheme over an arbitrary commutative ring, we prove that the unbounded derived category of quasi-coherent sheaves on X can be obtained from a category of Σ^op-diagrams of chain complexes of modules by inverting maps which induce homology isomorphisms on hyper-derived inverse limits. Moreover, we show that there is a finite set of weak generators, one for each cone in the fan Σ. The approach taken uses the machinery of Bousfield–Hirschhorn colocalisation of model categories. The first step is to characterise colocal objects; these turn out to be homotopy sheaves in the sense that chain complexes over different open sets U _σ agree on intersections up to quasi-isomorphism. In a second step it is shown that the homotopy category of homotopy sheaves is equivalent to the derived category of X.     

Effective algebraic degeneracy

We show that for every smooth projective hypersurface X⊂ℙ ⁿ⁺¹ of degree d and of arbitrary dimension n ≥2, if X is generic, then there exists a proper algebraic subvariety Y ⊊ X such that every nonconstant entire holomorphic curve f :ℂ→X has image f(ℂ) which lies in Y, as soon as its degree satisfies the effective lower bound d\geqslant 2ⁿ⁵d\geqslant 2^{n^{5}} .     

Symmetric group actions on homotopy S 2 × S 2

Let X be a closed smooth 4-manifold which is homotopy equivalent to S ² × S ². In this paper we use Seiberg-Witten theory to prove that if X admits a spin symmetric group S ₄ action of even type with b ₂ ⁺ (X/S ₄) = b ₂ ⁺ (X), then as an element of R (S ₄), Ind _S4 D _X = k ₁ (1 − θ) + k ₂(ψ₁ − ψ₂) for some integers k ₁ and k ₂, where 1, θ, ψ₁, ψ₂ are irreducible characters of S ₄ of degree 1, 1, 3, and 3 respectively. Authors’ address: Ximin Liu and Hongxia Li, Department of Applied Mathematics, Dalian University of Technology, Dalian 116024, P.R. China     

S-forcing,I. A “black-box” theorem for morasses,with applications to super-Souslin trees

We formulate, for regular μ>ω, a “forcing principle” S_μ which we show is equivalent to the existence of morasses, thus providing a new and systematic method for obtaining applications of morasses. Various examples are given, notably that for infinitek, if 2 ^k =k ⁺ and there exists a (k ⁺, 1)-morass, then there exists ak ⁺⁺-super-Souslin tree: a normalk ⁺⁺ tree characterized by a highly absolute “positive” property, and which has ak ⁺⁺-Souslin subtree. As a consequence we show that CH+SH_{ℵ 2}⟹ℵ₂ is (inaccessible)^L. This author thanks the US-Israel Binational Science Foundation for partial support of this research.     

Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads

In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts.The numbers N(f,k;X−{Xν}_ν∈M) and N(f,k;X,{Xν}_ν∈M), generalize the generalizations to coincidences, of Zhao's Nielsen number on the complement N(f;X−A), respectively Schirmer's relative Nielsen number N(f;X,A). Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps f^′ and k^′ which are homotopic as maps ofM-ads to f, respectively k firstly on the complement of the union of the subspaces Xν in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer's triad Nielsen number N(f;X₁∪X₂).Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here.     

Measure concentration for a class of random processes

Summary.   Let X={X _i } _{i =−∞} ^∞ be a stationary random process with a countable alphabet and distribution q. Let q ^∞(·|x _{− k} ⁰) denote the conditional distribution of X ^∞=(X ₁,X ₂,…,X _n ,…) given the k-length past:

Confined subgroups in periodic simple finitary linear groups

A subgroupX of the locally finite groupG is said to beconfined, if there exists a finite subgroupF≤G such thatX ^g∩F≠1 for allg∈G. Since there seems to be a certain correspondence between proper confined subgroups inG and non-trivial ideals in the complex group algebra ℂG, we determine the confined subgroups of periodic simple finitary linear groups in this paper. Dedicated to the memory of our friend and collaborator Richard E. Phillips     

Uniqueness theorem of solutions for stochastic differential equation in the plane

LetM={M _z, z ∈ R ₊ ² } be a continuous square integrable martingale andA={A _z, z ∈ R ₊ ² be a continuous adapted increasing process. Consider the following stochastic partial differential equations in the plane:dX _z=α(z, X_z)dM_z+β(z, X_z)dA_z, z∈R ₊ ² , X_z=Z_z, z∈∂R ₊ ² , whereR ₊ ² =[0, +∞)×[0,+∞) and ∂R ₊ ² is its boundary,Z is a continuous stochastic process on ∂R ₊ ² . We establish a new theorem on the pathwise uniqueness of solutions for the equation under a weaker condition than the Lipschitz one. The result concerning the one-parameter analogue of the problem we consider here is immediate (see [1, Theorem 3.2]). Unfortunately, the situation is much more complicated for two-parameter process and we believe that our result is the first one of its kind and is interesting in itself. We have proved the existence theorem for the equation in [2]. Supported by the National Science Foundation and the Postdoctoral Science Foundation of China     

CR Hypersurfaces with a Contracting Automorphism

Let M be a real analytic CR hypersurface in ℂ ⁿ⁺¹ admitting no varieties of positive dimension. We show first that every contracting local CR automorphism of M is linearizable. As a consequence, we show that such M admitting a contracting local CR automorphism is holomorphically equivalent to a weighted homogeneous hypersurface. Finally, we apply these results to prove that a bounded domain in ℂ ⁿ⁺¹ with a real analytic boundary admitting an automorphism contracting at a boundary point must admit a Lie subgroup of real dimension at least two in its automorphism group. Research of the first named author is partially supported by The Grant R01-2005-000-10771-0 of The Korea Science and Engineering Foundation.     

On Projecting Symmetries by Unbranched Regular Coverings of Riemann Surfaces

A symmetry of a Riemann surface X of genus g is an antiholomorphic involution σ of X. It is a classical result of Harnack that the set of fixed points of σ consists of k closed Jordan curves, called ovals, for some k, 0 ≤ k ≤ g + 1; when k = g or k = g+1 we say, following Natanzon [8], that σ is an (M – 1)- or an M-symmetry, respectively. Given a Riemann surface X with an M-symmetry, a Riemann surface Y and a regular covering p: X → Y, we prove that Y admits either an M- or an (M – 1)-symmetry and whenever p is unbranched we describe the groups of covering transformations of p. In the case that X is hyperelliptic we calculate as well the number of unbranched regular coverings p: X → Y in which X has an M-symmetry. The first two authors are supported by MTM2005-01637, the third by SAB2005-0049.