Symplectic <Emphasis Type="Italic">S</Emphasis><Subscript>5</Subscript> action on symplectic homotopy K3 surfaces

Let X be a symplectic homotopy K3 surface and G = S ₅ act on X symplectically. In this paper, we give a weak classification of the G action on X by discussing the fixed-point set structure. Besides, we analyse the exoticness of smooth structures of X under the action of G.     

Relative symplectic caps, 4-genus and fibered knots

We prove relative versions of the symplectic capping theorem and sufficiency of Giroux’s criterion for Stein fillability and use these to study the 4-genus of knots. More precisely, suppose we have a symplectic 4-manifold X with convex boundary and a symplectic surface Σ in X such that ?Σ is a transverse knot in ?X. In this paper, we prove that there is a closed symplectic 4-manifold Y with a closed symplectic surface S such that (X,Σ) embeds into (Y,S) symplectically. As a consequence we obtain a relative version of the symplectic Thom conjecture. We also prove a relative version of the sufficiency part of Giroux’s criterion for Stein fillability, namely, we show that a fibered knot whose mondoromy is a product of positive Dehn twists bounds a symplectic surface in a Stein filling. We use this to study 4-genus of fibered knots in \(\mathbb {S}^{3} \). Further, we give a criterion for quasipositive fibered knots to be strongly quasipositive.     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

Random difference equations with subexponential innovations

We consider the random difference equations S = _d (X + S)Y and T = _d X + TY, where = _d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.     

Asymptotic <Emphasis Type="Italic">T</Emphasis><Subscript><Emphasis Type="Italic">u</Emphasis></Subscript>-Toeplitzness of weighted composition operators on <Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>

Given a unilateral forward shift S acting on a complex, separable, innite dimensional Hilbert space H, an asymptotically S-Toeplitz operator is a bounded linear operator T on H satisfying that {S* ⁿ TS ⁿ } is convergent with respect to one of the topologies commonly used in the algebra of bounded linear operators on H. In this paper, we study the asymptotic T _u -Toeplitzness of weighted composition operators on the Hardy space H², where u is a nonconstant inner function.     

Holomorphic vector bundles on open subsets of Stein manifolds

LetM be a connected two-dimensional Stein manifold withH ²(M,Z)=0 andS∈M a discrete subset withS≠ Ø. SetX:=M/S. Fix an integerr≥2. Then there exists a rankr vector bundleE onX such that there is no line bundleL onX with a non-zero mapL →E.     

Representation and Approximation of Positivity Preservers

We consider a closed set S?? ⁿ and a linear operator

$\Phi \colon \mathbb{R}[X_1,\ldots,X_n]\rightarrow \mathbb{R}[X_1,\ldots,X_n]$

that preserves nonnegative polynomials, in the following sense: if f≥0 on S, then Φ(f)≥0 on S as well. We show that each such operator is given by integration with respect to a measure taking nonnegative functions as its values. This can be seen as a generalization of Haviland’s Theorem, which concerns linear functionals on ?[X ₁,…,X _n ]. For compact sets S we use the result to show that any nonnegativity preserving operator is a pointwise limit of very simple nonnegativity preservers with finite dimensional range.

     

Compactification of the space of branched coverings of the two-dimensional sphere

For a closed oriented surface Σ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let X_Σ,n be the set of isomorphism classes of orientation-preserving n-fold branched coverings Σ → S ² of the two-dimensional sphere. We complete X_Σ,n with the isomorphism classes of mappings that cover the sphere by the degenerations of Σ. In the case Σ = S ², the topology that we define on the obtained completion \({\overline X _{\Sigma ,n}}\) coincides on \({X_{{s^2},n}}\) with the topology induced by the space of coefficients of rational functions P/Q, where P and Q are homogeneous polynomials of degree n on ?P¹ ≌ S ². We prove that \({\overline X _{\Sigma ,n}}\) coincides with the Diaz–Edidin–Natanzon–Turaev compactification of the Hurwitz space H(Σ, n) ? X _Σ,n consisting of isomorphism classes of branched coverings with all critical values being simple.     

Semipositivity of generic ordinary fibration over a curve

Let f : X → B be a generic ordinary proper fibration over a complete curve in positive characteristics, we prove that the dual of higher direct image sheaf R~1 f_*O_X is nef. As a corollary, we show that f_*ω_(S/B) is nef, if f : S → B is a fibration from a surface to a curve with generic ordinary fibres.     

On the extension of Hölder maps with values in spaces of continuous functions

We study the isometric extension problem for Hölder maps from subsets of any Banach space intoc ₀ or into a space of continuous functions. For a Banach spaceX, we prove that anyα-Hölder map, with 0<α ≤1, from a subset ofX intoc ₀ can be isometrically extended toX if and only ifX is finite dimensional. For a finite dimensional normed spaceX and for a compact metric spaceK, we prove that the set ofα’s for which allα-Hölder maps from a subset ofX intoC(K) can be extended isometrically is either (0, 1] or (0, 1) and we give examples of both occurrences. We also prove that for any metric spaceX, the above described set ofα’s does not depend onK, but only on finiteness ofK.     

A Cantor set in the plane and its monotone subsets

Given c >  0 a planar Cantor set X with a dim _H (X) < 2 is constructed such that each c-monotone subspace of X has a smaller Hausdorff dimension than X.     

On the Chow groups of certain cubic fourfolds

This note is about the Chow groups of a certain family of smooth cubic fourfolds. This family is characterized by the property that each cubic fourfold X in the family has an involution such that the induced involution on the Fano variety F of lines in X is symplectic and has a K3 surface S in the fixed locus. The main result establishes a relation between X and S on the level of Chow motives. As a consequence, we can prove finite-dimensionality of the motive of certain members of the family.     

1-Cocycles on the group of contactomorphisms on the supercircle S1|3 generalizing the Schwarzian derivative

The relative cohomology H_diff¹(K(1|3), osp(2, 3);D_γ,µ(S^1|3)) of the contact Lie superalgebra K(1|3) with coefficients in the space of differential operators D_γ,µ(S^1|3) acting on tensor densities on S^1|3, is calculated in N.Ben Fraj, I. Laraied, S. Omri (2013) and the generating 1-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative 1-cocycle s(X_f) = D₁D₂D₃(_f)α₃^1/2, X^f ∈ K(1|3) which is invariant with respect to the conformal subsuperalgebra osp(2, 3) of K(1|3).     

Remarks on Ramanujam-Kawamata-Viehweg Vanishing Theorem

In this article we prove a general result on a nef vector bundle E on a projective manifold X of dimension n depending on the vector space H^n,n(X,E): It is also shown that H^n,n(X,E) = 0 for an indecomposable nef rank 2 vector bundles E on some specific type of n dimensional projective manifold X. The same vanishing shown to hold for indecomposable nef and big rank 2 vector bundles on any variety with trivial canonical bundle.     

Strange products of projections

Let H be an infinite-dimensional Hilbert space. We show that there exist three orthogonal projections X ₁,X ₂,X ₃ onto closed subspaces of H such that for every 0 ≠ z ₀ ∈ H there exist k ₁, k ₂, · · · ∈ {1, 2, 3} so that the sequence of iterates defined by z _n = X _kn z _n ?1 does not converge in norm.     

The Tukey Order and Subsets of <Emphasis Type="Italic">ω</Emphasis><Subscript>1</Subscript>

One partially ordered set, Q, is a Tukey quotient of another, P, if there is a map ? : P → Q carrying cofinal sets of P to cofinal sets of Q. Two partial orders which are mutual Tukey quotients are said to be Tukey equivalent. Let X be a space and denote by \(\mathcal {K}(X)\) the set of compact subsets of X, ordered by inclusion. The principal object of this paper is to analyze the Tukey equivalence classes of \(\mathcal {K}(S)\) corresponding to various subspaces S of ω ₁, their Tukey invariants, and hence the Tukey relations between them. It is shown that ω ^ω is a strict Tukey quotient of \({\Sigma }(\omega ^{\omega _{1}})\) and thus we distinguish between two Tukey classes out of Isbell’s ten partially ordered sets from (Isbell, J. R.: J. London Math Society 4(2), 394–416, 1972). The relationships between Tukey equivalence classes of \(\mathcal {K}(S)\), where S is a subspace of ω ₁, and \(\mathcal {K}(M)\), where M is a separable metrizable space, are revealed. Applications are given to function spaces.     

On the Győry-Sárközy-Stewart conjecture in function fields

We consider function field analogues of the conjecture of Gy?ry, Sárközy and Stewart (1996) on the greatest prime divisor of the product (ab+1)(ac+1)(bc+1) for distinct positive integers a, b and c. In particular, we show that, under some natural conditions on rational functions F,G,H ∈ ?(X), the number of distinct zeros and poles of the shifted products FH +1 and GH +1 grows linearly with degH if degH > max{deg F, degG}. We also obtain a version of this result for rational functions over a finite field.     

Affine ADE bundles over surfaces with $$ p _{g}=0$$

Given any Kodaira curve C in a complex surface X, we construct a simply-laced affine Lie algebra bundle \(\mathcal {E}\) over X. When \( p _{g}(X)=0\), we construct deformations of holomorphic structures on \(\mathcal {E}\) such that the new bundle is trivial over any ADE curve \( C^{\prime }\) inside C and therefore descends to the singular surface obtained by contracting \(C^{\prime }\).     

Virtual Betti numbers and virtual symplecticity of 4-dimensional mapping tori, Ⅱ

We show that if the fiber of a closed 4-dimensional mapping torus X is reducible and not S~2×S~1 or RP~3#RP~3, then the virtual first Betti number of X is infinite and X is not virtually symplectic. This confirms two conjectures made by Li and Ni(2014) in an earlier paper.