On microfunctions at the boundary along CR manifolds

Let X be a complex analytic manifold, a C ² submanifold, an openset with C ² boundary .Denote by (resp. ) the microlocalization along M (resp. ) of the sheaf of holomorphic functions.In the literature (cf. [A-G], [K-S 1,2])one encounters two classical results concerning the vanishing of the cohomology groups .The most general gives the vanishing outside a range of indices j whose length is equal to (with being the number of respectively positive, negative and null eigenvalues for themicrolocal Levi form ).The sharpest result gives the concentration in a single degree, provided that the difference is locally constant for near p (with for z the base point of p).The first result was restated for the complex in [D'A-Z 2], in the case codim We extend it here to any codimension and moreover we also restate for the second vanishing theorem.We also point out that the principle of our proof, related to a criterion for constancy of sheaves due to [K-S 1], is a quite new one.     

Systèmes dynamiques non archimédiens et corps des norms (Non-Archimedean Dynamic Systems and Fields of Norms)

Let _k be the ring of integers of a finite extension k of the field _p of p-adic numbers. The endomorphisms of a formal group law defined over _k provide nontrivial examples of commuting formal series with coefficients in _k . This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in _p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over _p . We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of _p -extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing ebyev polynomials     

Sur la cohomologie à supports compacts des variétés de Shimura pour GSp(4)_\mathbb{Q}

In this paper we compute the cohomology with compact supports of a Siegelthreefold as a virtual module over the product of the Galois group of over and the Hecke algebra. We use a method which has been developed by Ihara, Langlands and Kottwitz: comparison of the Grothendieck--Lefschetz formula and the Arthur--Selberg trace formula.     

On the Irreducibility of Commuting Varieties Associated with Involutions of Simple Lie Algebras

Let be a reductive Lie algebra over an algebraically closed field of characteristic zero and an arbitrary -grading. We consider the variety , which is called the commuting variety associated with the -grading. Earlier it was proved by the author that is irreducible, if the -grading is of maximal rank. Now we show that is irreducible for and (E₆,F₄). In the case of symmetric pairs of rank one, we show that the number of irreducible components of is equal to that of nonzero non--regular nilpotent G ₀-orbits in . We also discuss a general problem of the irreducibility of commuting varieties.     

An Application of U(g)-bimodules to Representation Theory of Affine Lie Algebras

Let be the affine Lie algebra associated to the simple finite-dimensional Lie algebra . We consider the tensor product of the loop -module associated to the irreducible finite-dimensional -module V() and the irreducible highest weight -module L _k,. Then L _k, can be viewed as an irreducible module for the vertex operator algebra M _k,0. Let A(L _k,) be the corresponding -bimodule. We prove that if the -module is zero, then the -module is irreducible. As an example, we apply this result on integrable representations for affine Lie algebras.     

The Local Stark Conjecture at a Real Place

A refinement of the rank 1 Abelian Stark conjecture has been formulated by B.Gross. This conjecture predicts some -adic analytic nature of a modification of the Stark unit. The conjecture makes perfect sense even when is an Archimedean place. Here we consider the conjecture when is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the conjecture for CM extensions; here the original Stark conjecture is uninteresting, but the refined conjecture is nontrivial. In more generality, we show that, under mild hypotheses, if the subgroup of the Galois group generated by complex conjugations has less than full rank, then the refined conjecture implies that the Stark unit should be a square. This phenomenon has been discovered by Dummit and Hayes in a particular type of situation. We show that it should hold in much greater generality.     

On Complete Arcs Arising from Plane Curves

We point out an interplay between -Frobenius non-classical plane curves and complete -arcs in . A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete -arcs with parameters and being a power of the characteristic. In addition, for q a square, new complete -arcs with either and or and are constructed by using certain reducible plane curves.     

On the Karoubi Filtration of a Category

Let be an -filtered category in the sense of Karoubi. This is the categorical analogue of an ideal in a ring . Pedersen and Weibel constructed a fibration of K-theory spectra associated with the sequence . We present a new easier proof based on Waldhausen' generic fibration.     

Covariant Hyperelliptic Functions of Genus Two

We report results of investigations concerning the role of representations of in the theory of genus-two hyperelliptic functions. We discuss the role of these representations in the classical theory as well as introduce a family of new, naturally covariant functions.     

Lp Contraction Semigroups for Vector Valued Functions

Let be a contraction semigroup on the space of vector valued functions ( is a Hilbert space). In order to study the extension of to a contaction semigroup on , Shigekawa [Sh] studied recently the domination property where is a symmetric sub-Markovian semigroup on . He gives in the setting of square field operators sufficient conditions for the above inequality. The aim of the present paper is to show that the methods of [12] and [13] can be applied in the present setting and provide two ways for the extension of to We give necessary and sufficient conditions in terms of sesquilinear forms for the contractivity property as well as for the above domination property in a more general situation.     

Moduli Spaces of Stable Polygons and Symplectic Structures on\overline {\mathcal{M}} _{0,n}

In this paper, certain natural and elementary polygonal objects in Euclidean space, the stable polygons, are introduced, and the novel moduli spaces of stable polygons are constructed as complex analytic spaces. Quite unexpectedly, these new moduli spaces are shown to be projective and isomorphic to the moduli space of the Deligne–Mumford stable curves of genus 0. Further, built into the structures of stable polygons are some natural data giving rise to a family of (classes of) symplectic (Kähler) forms. This, via the link to , brings up a new tool to study the Kähler topology of . A wild but precise conjecture on the shape of the Kähler cone of is given in the end.     

Erdős–Szekeres-Type Theorems for Segments and Noncrossing Convex Sets

A family of convex sets is said to be in convex position, if none of its members is contained in the convex hull of the others. It is proved that there is a function N(n) with the following property. If is a family of at least N(n) plane convex sets with nonempty interiors, such that any two members of have at most two boundary points in common and any three are in convex position, then has n members in convex position. This result generalizes a theorem of T. Bisztriczky and G. Fejes Tóth. The statement does not remain true, if two members of may share four boundary points. This follows from the fact that there exist infinitely many straight-line segments such that any three are in convex position, but no four are. However, there is a function M(n) such that every family of at least M(n) segments, any four of which are in convex position, has n members in convex position.     

Hypersurfaces in ℝn and critical points in their external region

In this paper we study the hypersurfaces given as connected compact regular fibers of a differentiable map , in the cases in which has finitely many nondegenerate critical points in the unbounded component of .     

Determinants in Triangulated Categories

For a small triangulated category , Bass's K ₁ group is described, and the theorem of the heart is proved. We define the determinant map from to Neeman's , and we compute this map when is the derived category of an Abelian category .     

Formulas for Lagranigian and orthogonal degeneracy loci; \widetilde Q-polynomial approach

The main goal of the paper is to give explicit formulas for the fundamental classes of Schubert subschemes in Lagrangian and orthogonal Grassmannians of maximal isotropic subbundles as well as some globalizations of them. The used geometric tools overlap appropriate desingularizations of such Schubert subschemes and Gysin maps for such Grassmannian bundles. The main algebraic tools are provided by the families of and -polynomials introduced and investigated in the present paper. The key technical result of the paper is the computation of the class of the (relative) diagonal in isotropic Grassmannian bundles based on the orthogonality property of and polynomials. Some relationships with quaternionic Schubert varieties and Schubert polynomials for classical groups are also discussed.     

Orbits of Curves on Certain K3 Surfaces

In this paper, we study the family of algebraic K3 surfaces generated by the smooth intersection of a (1, 1) form and a (2, 2) form in defined over and with Picard number 3. We describe the group of automorphisms on V. For an ample divisor D and an arbitrary curve C ₀ on V, we investigate the asymptotic behavior of the quantity . We show that the limit

A Question in the Theory of Totally Local Formations of Finite Groups

It is proved that all proper totally local subformations of a non one-generated totally local formation of finite groups are one-generated iff coincides with a formation of all soluble -groups, where ||=2.     

Commutativity of the Arens product in lattice ordered algebras

Let be an Abelian Archimedean lattice ordered algebra. The order bidual furnished with the Arens product is again a lattice ordered algebra. We show that the order continuous order bidual is Abelian. This solves an open problem and improves a result of Scheffold, who proved it for the case of normed lattice ordered algebras. The proof is based on the up-down-up approximation of positive elements in the order continuous order bidual by elements in the canonical image of in Components of positive elements in are characterized and the result is applied to the Arens product of -and almost -algebras.