Sets of uniqueness for spherically convergent multiple trigonometric series

A subset of the -dimensional torus is called a set of uniqueness, or -set, if every multiple trigonometric series spherically converging to outside vanishes identically. We show that all countable sets are -sets and also that sets are -sets for every . In particular, , where is the Cantor set, is an set and hence a -set. We will say that is a -set if every multiple trigonometric series spherically Abel summable to outside and having certain growth restrictions on its coefficients vanishes identically. The above-mentioned results hold also for sets. In addition, every -set has measure , and a countable union of closed -sets is a -set.

     

Coloring

If and , then define the graph to be the graph whose vertex set is with two vertices being adjacent iff there are distinct such that . For various and and various , typically or , the graph can be properly colored with colors. It is shown that in some cases such a coloring can also have the additional property that if is an isometric embedding, then the restriction of to is a bijection onto .

     

Milnor classes of local complete intersections

Let be a compact local complete intersection defined as the zero set of a section of a holomorphic vector bundle over the ambient space. For each connected component of the singular set of , we define the Milnor class in the homology of . The difference between the Schwartz-MacPherson class and the Fulton-Johnson class of is shown to be equal to the sum of over the connected components of . This is done by proving Poincaré-Hopf type theorems for these classes with respect to suitable tangent frames. The -degree component coincides with the Milnor numbers already defined by various authors in particular situations. We also give an explicit formula for when is a non-singular component and satisfies the Whitney condition along .

     

Extensions for finite Chevalley groups II

Let be a semisimple simply connected algebraic group defined and split over the field with elements, let be the finite Chevalley group consisting of the -rational points of where , and let be the th Frobenius kernel. The purpose of this paper is to relate extensions between modules in and with extensions between modules in . Among the results obtained are the following: for 2$"> and , the -extensions between two simple -modules are isomorphic to the -extensions between two simple -restricted -modules with suitably ``twisted" highest weights. For , we provide a complete characterization of where and is -restricted. Furthermore, for , necessary and sufficient bounds on the size of the highest weight of a -module are given to insure that the restriction map is an isomorphism. Finally, it is shown that the extensions between two simple -restricted -modules coincide in all three categories provided the highest weights are ``close" together.

     

Existence of curves with prescribed topological singularities

Throughout this paper we study the existence of irreducible curves on smooth projective surfaces with singular points of prescribed topological types . There are necessary conditions for the existence of the type for some fixed divisor on and suitable coefficients , and , and the main sufficient condition that we find is of the same type, saying it is asymptotically proper. Ten years ago general results of this quality were not known even for the case . An important ingredient for the proof is a vanishing theorem for invertible sheaves on the blown up of the form , deduced from the Kawamata-Vieweg Vanishing Theorem. Its proof covers the first part of the paper, while the middle part is devoted to the existence theorems. In the last part we investigate our conditions on ruled surfaces, products of elliptic curves, surfaces in , and K3-surfaces.

     

Witten-Helffer-Sjöstrand theory for -equivariant cohomology

Given an -invariant Morse function and an -invariant Riemannian metric , a family of finite dimensional subcomplexes , , of the Witten deformation of the -equivariant de Rham complex is constructed, by studying the asymptotic behavior of the spectrum of the corresponding Laplacian as . In fact the spectrum of can be separated into the small eigenvalues, finite eigenvalues and the large eigenvalues. Then one obtains as the complex of eigenforms corresponding to the small eigenvalues of . This permits us to verify the -equivariant Morse inequalities. Moreover suppose is self-indexing and satisfies the Morse-Smale condition, then it is shown that this family of subcomplexes converges as to a geometric complex which is induced by and calculates the -equivariant cohomology of .

     

Topological mixing in -spaces

If is a proper -space and a non-elementary discrete group of isometries acting properly discontinuously on it is shown that the geodesic flow on the quotient space is topologically mixing, provided that the generalized Busemann function has zeros on the boundary and the non-wandering set of the flow equals the whole quotient space of geodesics (the latter being redundant when is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete -spaces by a one-ended group of isometries and (C) finite -dimensional ideal polyhedra.

     

Generalized space forms

Spaces with radially symmetric curvature at base point are shown to be diffeomorphic to space forms. Furthermore, they are either isometric to or under a radially symmetric metric, to with Riemannian universal covering of equipped with a radially symmetric metric, or else have constant curvature outside a metric ball of radius equal to the injectivity radius at .

     

Finely -harmonic functions of a Markov process

Let be a Borel right process and a fixed excessive measure. Given a finely open nearly Borel set we define an operator which we regard as an extension of the restriction to of the generator of . It maps functions on to (locally) signed measures on not charging -semipolars. Given a locally smooth signed measure we define to be (finely) -harmonic on provided on and denote the class of such by . Under mild conditions on we show that is equivalent to a local ``Poisson' representation of . We characterize by an analog of the mean value property under secondary assumptions. We obtain global Poisson type representations and study the Dirichlet problem for elements of under suitable finiteness hypotheses. The results take their nicest form when specialized to Hunt processes.

     

Tight closure, plus closure and Frobenius closure in cubical cones

We consider tight closure, plus closure and Frobenius closure in the rings , where is a field of characteristic and . We use a -grading of these rings to reduce questions about ideals in the quotient rings to questions about ideals in the regular ring . We show that Frobenius closure is the same as tight closure in certain classes of ideals when . Since , we conclude that for these ideals. Using injective modules over the ring , the union of all th roots of elements of , we reduce the question of whether for -graded ideals to the case of -graded irreducible modules. We classify the irreducible -primary -graded ideals. We then show that for most irreducible -primary -graded ideals in , where is a field of characteristic and . Hence for these ideals.

     

Induced operators on symmetry classes of tensors

Let be an -dimensional Hilbert space. Suppose is a subgroup of the symmetric group of degree , and is a character of degree 1 on . Consider the symmetrizer on the tensor space

defined by and . The vector space

is a subspace of , called the symmetry class of tensors over associated with and . The elements in of the form are called decomposable tensors and are denoted by . For any linear operator acting on , there is a (unique) induced operator acting on satisfying

In this paper, several basic problems on induced operators are studied.

     

Frobenius morphisms and representations of algebras

By introducing Frobenius morphisms on algebras and their modules over the algebraic closure of the finite field of elements, we establish a relation between the representation theory of over and that of the -fixed point algebra over . More precisely, we prove that the category    mod- of finite-dimensional -modules is equivalent to the subcategory of finite-dimensional -stable -modules, and, when is finite dimensional, we establish a bijection between the isoclasses of indecomposable -modules and the -orbits of the isoclasses of indecomposable -modules. Applying the theory to representations of quivers with automorphisms, we show that representations of a modulated quiver (or a species) over can be interpreted as -stable representations of the corresponding quiver over . We further prove that every finite-dimensional hereditary algebra over is Morita equivalent to some , where is the path algebra of a quiver over and is induced from a certain automorphism of . A close relation between the Auslander-Reiten theories for and is established. In particular, we prove that the Auslander-Reiten (modulated) quiver of is obtained by ``folding" the Auslander-Reiten quiver of . Finally, by taking Frobenius fixed points, we are able to count the number of indecomposable representations of a modulated quiver over with a given dimension vector and to generalize Kac's theorem for all modulated quivers and their associated Kac-Moody algebras defined by symmetrizable generalized Cartan matrices.

     

Degenerate fibres in the Stone-Cech compactification of the universal bundle of a finite group

Applied to a continuous surjection of completely regular Hausdorff spaces and , the Stone-Cech compactification functor yields a surjection . For an -fold covering map , we show that the fibres of , while never containing more than points, may degenerate to sets of cardinality properly dividing . In the special case of the universal bundle of a -group , we show more precisely that every possible type of -orbit occurs among the fibres of . To prove this, we use a weak form of the so-called generalized Sullivan conjecture.

     

Block representation type of reduced enveloping algebras

Let be an algebraically closed field of characteristic , a connected, reductive -group, , and the reduced enveloping algebra of associated with . Assume that is simply-connected, is good for and has a non-degenerate -invariant bilinear form. All blocks of having finite and tame representation type are determined.

     

Polynomials nonnegative on a grid and discrete optimization

We characterize the real-valued polynomials on that are nonnegative (not necessarily strictly positive) on a grid of points of , in terms of a weighted sum of squares whose degree is bounded and known in advance. We also show that the mimimization of an arbitrary polynomial on (a discrete optimization problem) reduces to a convex continuous optimization problem of fixed size. The case of concave polynomials is also investigated. The proof is based on a recent result of Curto and Fialkow on the -moment problem.

     

Sufficient conditions for zero-one laws

We generalize a result of Bateman and Erdos concerning partitions, thereby answering a question of Compton. From this result it follows that if is a class of finite relational structures that is closed under the formation of disjoint unions and the extraction of components, and if it has the property that the number of indecomposables of size is bounded above by a polynomial in , then has a monadic second order - law. Moreover, we show that if a class of finite structures with the unique factorization property is closed under the formation of direct products and the extraction of indecomposable factors, and if it has the property that the number of indecomposables of size at most is bounded above by a polynomial in , then this class has a first order - law. These results cover all known natural examples of classes of structures that have been proved to have a logical - law by Compton's method of analyzing generating functions.

     

Uniform and Lipschitz homotopy classes of maps

If is a compact connected polyhedron, we associate with each uniform homotopy class of uniformly continuous mappings from the real line into an element of where is the space of uniformly continuous functions from to and is the subspace of bounded uniformly continuous functions. This map from uniform homotopy classes of functions to is surjective. If is the -dimensional torus, it is bijective, while if is a compact orientable surface of genus 1$">, it is not injective.

In higher dimensions we have to consider smooth Lipschitz homotopy classes of smooth Lipschitz maps from suitable Riemannian manifolds to compact smooth manifolds With each such Lipschitz homotopy class we associate an element of where is the dimension of is the space of bounded continuous functions from the positive real axis to and is the set of all such that

     

Large orbits in coprime actions of solvable groups

Let be a solvable group of automorphisms of a finite group . If and are coprime, then there exists an orbit of on of size at least . It is also proved that in a -solvable group, the largest normal -subgroup is the intersection of at most three Hall -subgroups.

     

On the blow-up of heat flow for conformal -harmonic maps

Using a comparison theorem, Chang, Ding, and Ye (1992) proved a finite time derivative blow-up for the heat flow of harmonic maps from (a unit ball in ) to (a unit sphere in ) under certain initial and boundary conditions. We generalize this result to the case of -harmonic map heat flow from to . In contrast to the previous case, our governing parabolic equation is quasilinear and degenerate. Technical issues such as the development of a new comparison theorem have to be resolved.

     

On Fano manifolds with Nef tangent bundles admitting 1-dimensional varieties of minimal rational tangents

Let be a Fano manifold of Picard number 1 with numerically effective tangent bundle. According to the principal case of a conjecture of Campana-Peternell's, should be biholomorphic to a rational homogeneous manifold , where is a simple Lie group, and is a maximal parabolic subgroup.

In our opinion there is no overriding evidence for the Campana-Peternell Conjecture for the case of Picard number 1 to be valid in its full generality. As part of a general programme that the author has undertaken with Jun-Muk Hwang to study uniruled projective manifolds via their varieties of minimal rational tangents, a new geometric approach is adopted in the current article in a special case, consisting of (a) recovering the generic variety of minimal rational tangents , and (b) recovering the structure of a rational homogeneous manifold from . The author proves that, when and the generic variety of minimal rational tangents is 1-dimensional, is biholomorphic to the projective plane , the 3-dimensional hyperquadric , or the 5-dimensional Fano homogeneous contact manifold of type , to be denoted by .

The principal difficulty is part (a) of the scheme. We prove that is a rational curve of degrees , and show that resp. 2 resp. 3 corresponds precisely to the cases of resp. resp. . Let be the normalization of a choice of a Chow component of minimal rational curves on . Nefness of the tangent bundle implies that is smooth. Furthermore, it implies that at any point , the normalization of the corresponding Chow space of minimal rational curves marked at is smooth. After proving that is a rational curve, our principal object of study is the universal family of , giving a double fibration , which gives -bundles. There is a rank-2 holomorphic vector bundle on whose projectivization is isomorphic to . We prove that is stable, and deduce the inequality from the inequality resulting from stability and the existence theorem on Hermitian-Einstein metrics. The case of is ruled out by studying the structure of the curvature tensor of the Hermitian-Einstein metric on in the special case where .