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 .

     

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.

     

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.

     

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.

     

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.

     

Weakly defective varieties

A projective variety is `-weakly defective' when its intersection with a general -tangent hyperplane has no isolated singularities at the points of tangency. If is -defective, i.e. if the -secant variety of has dimension smaller than expected, then is also -weakly defective. The converse does not hold in general. A classification of weakly defective varieties seems to be a basic step in the study of defective varieties of higher dimension. We start this classification here, describing all weakly defective irreducible surfaces. Our method also provides a new proof of the classical Terracini's classification of -defective surfaces.

     

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.

     

Sums of squares in real analytic rings

Let be an analytic ring. We show: (1) has finite Pythagoras number if and only if its real dimension is , and (2) if every positive semidefinite element of is a sum of squares, then is real and has real dimension .

     

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.

     

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

     

The extraspecial case of the problem

Let be an extraspecial-type group and a faithful, absolutely irreducible -module, where is a finite field. Let be the normalizer in of . We show that, with few exceptions, there exists a such that the restriction of to is self-dual whenever and .

     

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.

     

Shellability in reductive monoids

The purpose of this paper is to extend to monoids the work of Björner, Wachs and Proctor on the shellability of the Bruhat-Chevalley order on Weyl groups. Let be a reductive monoid with unit group , Borel subgroup and Weyl group . We study the partially ordered set of -orbits (with respect to Zariski closure inclusion) within a -orbit of . This is the same as studying a -orbit in the Renner monoid . Such an orbit is the retract of a `universal orbit', which is shown to be lexicograhically shellable in the sense of Björner and Wachs.

     

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.

     

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 .

     

Small rational model of subspace complement

This paper concerns the rational cohomology ring of the complement of a complex subspace arrangement. We start with the De Concini-Procesi differential graded algebra that is a rational model for . Inside it we find a much smaller subalgebra quasi-isomorphic to the whole algebra. is described by defining a natural multiplication on a chain complex whose homology is the local homology of the intersection lattice whence connecting the De Concini-Procesi model with the Goresky-MacPherson formula for the additive structure of . The algebra has a natural integral version that is a good candidate for an integral model of . If the rational local homology of can be computed explicitly we obtain an explicit presentation of the ring . For example, this is done for the cases where is a geometric lattice and where is a -equal manifold.

     

Vertices for characters of -solvable groups

Suppose that is a finite -solvable group. We associate to every irreducible complex character of a canonical pair , where is a -subgroup of and , uniquely determined by up to -conjugacy. This pair behaves as a Green vertex and partitions into ``families" of characters. Using the pair , we give a canonical choice of a certain -radical subgroup of and a character associated to which was predicted by some conjecture of G. R. Robinson.

     

Rational -equivariant homotopy theory

We give an algebraicization of rational -equivariant homotopy theory. There is an algebraic category of `` -systems' which is equivalent to the homotopy category of rational -simply connected -spaces. There is also a theory of ``minimal models' for -systems, analogous to Sullivan's minimal algebras. Each -space has an associated minimal -system which encodes all of its rational homotopy information, including its rational equivariant cohomology and Postnikov decomposition.