The Grothendieck ring of varieties and piecewise isomorphisms

Let K ₀(Var _k ) be the Grothendieck ring of algebraic varieties over a field k. Let X, Y be two algebraic varieties over k which are piecewise isomorphic (i.e. X and Y admit finite partitions X ₁, ..., X _n , Y ₁, ..., Y _n into locally closed subvarieties such that X _i is isomorphic to Y _i for all i ≤ n), then [X] = [Y] in K ₀(Var _k ). Larsen and Lunts ask whether the converse is true. For characteristic zero and algebraically closed field k, we answer positively this question when dim X ≤ 1 or X is a smooth connected projective surface or if X contains only finitely many rational curves.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

Galois actions on torsion points of one-dimensional formal modules

Let F be a local non-Archimedean field with ring of integers o and uniformizer ?, and fix an algebraically closed extension k of the residue field of o. Let X be a one-dimensional formal o-module of F-height n over k. By the work of Drinfeld, the universal deformation X of X is a formal group over a power series ring R₀ in n−1 variables over Wo(k). For h∈{0,…,n−1} let Uh⊂Spec(R₀) be the locus where the connected part of the associated ?-divisible module X[?^∞] has height h. Using the theory of Drinfeld level structures we show that the representation of π₁(Uh) on the Tate module of the étale quotient is surjective.     

Splitting manifolds as M × R where M has a k-fold end structure

Let X be a locally finite simplicial complex of dimension n, n? 5, equipped with a k-fold end structure [4] and consider a piecewise linear (n + 1)-dimensional manifold M that is proper homotopy equivalent to X × R by F:M → X × R, where R is the set of real numbers. The question arises as to whether or not the manifold M can be split, i.e., written as M = N × R where N is a n-manifold and where there is a proper homotopy between F and (p₁ ° F₀) × id:N × R → X × R, preserving the natural (k+1)-fold end structure, where F₀ is F|_N and p₁ is the projection X × R → X. Of particular significance is the fact that X is noncompact. When the construction of such splittings is attempted, algebraic obstructions arise, which vanish if and only if the construction can be completed. This paper develops such an obstruction theory by utilizing methods of L.C. Siebenmann and the k-fold end structures of F. Waldhausen.     

Reduction of variance for Gaussian densities via restriction to convex sets

Let X be a random vector with values in $\text{R}$ⁿ and a Gaussian density f. Let Y be a random vector whose density can be factored as k · f, where k is a logarithmically concave function on $\text{R}$ⁿ. We prove that the covariance matrix of X dominates the covariance matrix of Y by a positive semidefinite matrix. When k is the indicator function of a compact convex set A of positive measure the difference is positive definite. If A and X are both symmetric Var(a · X) is bounded above by an expression which is always strictly less than Var(a · X) for every a ∈ $\text{R}$ⁿ. Finally some counterexamples are given to show that these results cannot be extended to the general case where f is any logarithmically concave density.     

Singularities of generic characteristic polynomials and smooth finite splittings of Azumaya algebras over surfaces

Let k be an algebraically closed field. Let P(X ₁₁, . . . , X _nn , T) be the characteristic polynomial of the generic matrix (X _ij ) over k. We determine its singular locus as well as the singular locus of its Galois splitting. If X is a smooth quasi-projective surface over k and A an Azumaya algebra on X of degree n, using a method suggested by M. Artin, we construct finite smooth splittings for A of degree n over X whose Galois closures are smooth.     

Euler–Poincaré formula in equal characteristic¶under ordinariness assumptions

Let X be an irreducible smooth projective curve over an algebraically closed field of characteristic p>0. Let ? be either a finite field of characteristic p or a local field of residue characteristic p. Let F be a constructible étale sheaf of $\BF$-vector spaces on X. Suppose that there exists a finite Galois covering π:Y→X such that the generic monodromy of π^* F is pro-p and Y is ordinary. Under these assumptions we derive an explicit formula for the Euler–Poincaré characteristic χ(X,F) in terms of easy local and global numerical invariants, much like the formula of Grothendieck–Ogg–Shafarevich in the case of different characteristic. Although the ordinariness assumption imposes severe restrictions on the local ramification of the covering π, it is satisfied in interesting cases such as Drinfeld modular curves. Received: 7 December 1999 / Revised version: 28 January 2000     

Waldhausen's theory of k-fold end structures: a survey

Let R⁺ be the space of nonnegative real numbers. F. Waldhausen defines a k-fold end structure on a space X as an ordered k-tuple of continuous maps x_f:X → R⁺, 1 ? j ? k, yielding a proper map x:X → (R⁺)^k. The pairs (X,x) are made into the category E^k of spaces with k-fold end structure. Attachments and expansions in E^k are defined by induction on k, where elementary attachments and expansions in E⁰ have their usual meaning. The category E^k/Z consists of objects (X, i) where i: Z → X is an inclusion in E^k with an attachment of i(Z) to X, and the category E^k6Z consists of pairs (X,i) of E^k/Z that admit retractions X → Z. An infinite complex over Z is a sequence X = {X₁ ? X₂ ? … ? X_n …} of inclusions in E^k6Z. The abelian grou p S₀(Z) is then defined as the set of equivalence classes of infinite complexes dominated by finite ones, where the equivalence relation is generated by homotopy equivalence and finite attachment; and the abelian group S₁(Z) is defined as the set of equivalence classes of X₁, where X ∈ E^k/Z deformation retracts to Z. The group operations are gluing over Z. This paper presents the Waldhausen theory with some additions and in particular the proof of Waldhausen's proposition that there exists a natural exact sequence 0 → S₁(Z × R)→^πS₀(Z) by utilizing methods of L.C. Siebenmann. Waldhausen developed this theory while seeking to prove the topological invariance of Whitehead torsion; however, the end structures also have application in studying the splitting of a noncompact manifold as a product with R[1].     

Real parabolic vector bundles over a real curve

We define real parabolic structures on real vector bundles over a real curve. Let (X, σ _X ) be a real curve, and let S???X be a non-empty finite subset of X such that σ _X (S)?=?S. Let N?≥?2 be an integer. We construct an N-fold cyclic cover p : Y→ X in the category of real curves, ramified precisely over each point of S, and with the property that for any element g of the Galois group Γ, and any y?∈?Y, one has $\sigma_Y(gy) = g^{-1}\sigma_Y(y)$ . We established an equivalence between the category of real parabolic vector bundles on (X, σ _X ) with real parabolic structure over S, all of whose weights are integral multiples of 1/N, and the category of real Γ-equivariant vector bundles on (Y, σ _Y ).     

On the fluctuations of independent copies of moving maxima

Let {X_n, n1} be a sequence of independent random variables (r.v.'s) with a common distribution function (d.f.) F. Define the moving maxima Y_k(n)=max(X_n−k(n)+1,X_n−k(n)+2,…,X_n), where {k(n), n1} is a sequence of positive integers. Let Y_k(n)¹ and Y_k(n)² be two independent copies of Y_k(n). Under certain conditions on F and k(n), the set of almost sure limit points of the vector consisting of properly normalised Y_k(n)¹ and Y_k(n)² is obtained.     

On the fundamental group-scheme

We show that the fundamental group-scheme of a separably rationally connected variety defined over an algebraically closed field is trivial. Let X be a geometrically irreducible smooth projective variety defined over a finite field k admitting a k-rational point. Let {En,σn}_n?0 be a flat principal G-bundle over X, where G is a reductive linear algebraic group defined over k. We show that there is a positive integer a such that the principal G-bundle is isomorphic to E₀, where FX is the absolute Frobenius morphism of X. From this it follows that E₀ is given by a representation of the fundamental group-scheme of X in G.     

On extensions of polynomial functions

Let X, Y be two linear spaces over the field ? of rationals and let D ≠ ? be a (?—convex subset of X. We show that every function ?: D → Y satisfying the functional equation $${\mathop\sum^{n+1}\limits_{j=0}}(-1)^{n+1-j}\Bigg(^{n+1}_{j}\Bigg)f\Bigg((1-{j\over {n+1}})x+{j\over{n+1}}y\Bigg)=0,\ \ \ x,y\in\ D,$$ admits an extension to a function F: X → Y of the form $$F(x)=A^o+A^1(x)+\cdot\cdot\cdot+A^n(x),\ \ \ x\in\ X,$$ where A ^o ∈ Y, A^k(x) ? A_k(x,…,x), x ∈ X, and the maps A _k: X ^k → Y are k—additive and symmetric, k ∈ {1,…, n}. Uniqueness of the extension is also discussed.     

Generalizations of a lemma of Freudenburg

Let k be an algebraically closed field of characteristic zero and ℘ a prime ideal in k[X]?k[X₁,…,X_n]. Let g∈k[X] and d?1. If for all 1?|α|?d the derivatives ∂^αg belong to ℘, then there exists c∈k such that g−c∈℘(d+1), the d+1th symbolic power of ℘. In particular, if ℘ is a complete intersection it follows that g−c∈℘d+1.     

A relative Shafarevich theorem

Suppose given a Galois étale cover YX of proper non-singular curves over an algebraically closed field k of prime characteristic p. Let H be its Galois group, and G any finite extension of H by a p-group P. We give necessary and sufficient conditions on G to be the Galois group of an étale cover of X dominating YX.in final form: 16 September 2003     

Nikol’skii’s inequality for different metrics and properties of the sequence of norms of the Fourier sums of a function in the Lorentz space

Let (X, Y) be a pair of normed spaces such that X ? Y ? L ₁[0, 1] ⁿ and {e _k } _k be an expanding sequence of finite sets in ? ⁿ with respect to a scalar or vector parameter k, k ∈ ? or k ∈ ? ⁿ . The properties of the sequence of norms $\{ \left\| {S_{e_k } (f)} \right\|x\} _k $ of the Fourier sums of a fixed function f ∈ Y are studied. As the spaces X and Y, the Lebesgue spaces L _p [0, 1], the Lorentz spaces L _p,q [0, 1], L _p,q [0, 1] ⁿ , and the anisotropic Lorentz spaces L _p,q*[0, 1] ⁿ are considered. In the one-dimensional case, the sequence {e _k } _k consists of segments, and in the multidimensional case, it is a sequence of hyperbolic crosses or parallelepipeds in ? ⁿ . For trigonometric polynomials with the spectrum given by step hyperbolic crosses and parallelepipeds, various types of inequalities for different metrics in the Lorentz spaces L _p,q [0, 1] ⁿ and L _p,q*[0, 1] ⁿ are obtained.     

Some generalizations of the first Fredholm theorem to multivalued A-proper mappings with applications to nonlinear elliptic equations

Let X and Y be real normed spaces with an admissible scheme Γ = {E_n, V_n; F_n, W_n} and T: X → 2^YA-proper with respect to Γ such that dist(y, A(x)) < kc(∥ x ∥) for all y in T(x) with ∥ x ∥ ? R for some R > 0 and k > 0, where c: R⁺ → R⁺ is a given function and A: X → 2^Y a suitable possibly not A-proper mapping. Under the assumption that either T or A is odd or that (u, Kx) ? 0 for all u in T(x) with $\text{∥ x ∥ ? r > 0}\text{and some}\text{K: X → Y}{}^1$, we obtain (in a constructive way) various generalizations of the first Fredholm theorem. The unique approximation-solvability results for the equation T(x) = f with T such that T(x) ? T(y) ?A(x ? y) for x, y in X or T is Fréchet differentiable are also established. The abstract results for A-proper mappings are then applied to the (constructive) solvability of some boundary value problems for quasilinear elliptic equations. Some of our results include the results of Lasota, Lasota-Opial, Hess, Ne?as, Petryshyn, and Babu?ka.     

The variety of the asymptotic values of a real polynomial etale map

A polynomial map F: R² → R² is said to satisfy the Jacobian condition if ∀(X, Y)ϵ R², J(F)(X, Y) ≠ 0. The real Jacobian conjecture was the assertion that such a map is a global diffeomorphism. Recently the conjecture was shown to be false by S. Pinchuk. According to a theorem of J. Hadamard any counterexample to the conjecture must have asymptotic values. We give the structure of the variety of all the asymptotic values of a polynomial map F: R² → R² that satisfies the Jacobian condition. We prove that the study of the asymptotic values of such maps can be reduced to those maps that have only X- or Y-finite asymptotic values. We prove that a Y-finite asymptotic value can be realized by F along a rational curve of the type (X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N), where X → 0, Y is fixed and K, N > 0 are integers. More precisely we prove that the coordinate polynomials P(U, V) of F(U, V) satisfy finitely many asymptotic identities, namely, identities of the following type, P(X^{− k}, A₀ + A₁ X + … + A_{N − 1} X^{N − 1} + YX^N) = A(X, Y)ϵ R[X, Y], which ‘capture’ the whole set of asymptotic values of F.     

Cyclicity of the unramified Iwasawa module

For a number field k and a prime number p, let k _?? be the cyclotomic Z _p -extension of k with finite layers k _n . We study the finiteness of the Galois group X _?? over k _?? of the maximal abelian unramified p-extension of k _?? when it is assumed to be cyclic. We then focus our attention to the case where p?=?2 and k is a real quadratic field and give the rank of the 2-primary part of the class group of k _n . As a consequence, we determine the complete list of real quadratic number fields for which X _?? is cyclic non trivial. We then apply these results to the study of Greenberg??s conjecture for infinite families of real quadratic fields thus generalizing previous results obtained by Ozaki and Taya.     

THE STRUCTURE OF THE TANGENT CONE: AN INTERESTING BIJECTION#

ABSTRACT

Let X = Spec(R) be a reduced equidimensional algebraic variety over an algebraically closed field k. Let Y = Spec(R/𝔮) be a codimension one ordinary multiple subvariety, where 𝔮 is a prime ideal of height 1 of R. If U is a nonempty open subset of Y and 𝔪 a closed point of U, we denote by A ? R _𝔪 its local ring in X, by 𝔭 the extension of 𝔮 in A, and by K the algebraic closure of the residue field k(𝔭).

Then there exists a bijection γ^𝔪:Proj(G _𝔭(A) ?  _A/𝔭 k) → Proj(G(A _𝔭) ?  _k(𝔭)K) such that for every subset Σ of Proj(G _𝔭(A) ?  _A/𝔭 k), the Hilbert function of Σ coincides with the Hilbert function of γ^𝔪(Σ). We examine some applications. We study the structure of the tangent cone at a closed point of a codimension one ordinary multiple subvariety.     

Strongly consistent estimators of k-th order regression curves and rates of convergence

Summary Let X and Y be two jointly distributed real valued random variables, and let the conditional distribution of X given Y be either in a Lebesgue exponential family or in a discrete exponential family. Let r_k be the k-th order regression curve of Y on X. Let X _n=(X ₁,..., X_n) be a random sample of size n on X. For a subset S of the real line R, statistics based on X_n are exhibited and sufficient conditions are given under which is close to O(n ^–1/2) with probability one. To obtain this result, with uf (u known and f unknown) denoting the unconditional (on y) density of X, the problem of estimating r _k (·) is reduced to the one of estimating f ^(k) (·)/f(·) if the density is wrt the Lebesgue measure on R and f ^(k) is the k-th order derivative of f; and to the one of estimating f(·+k)/f(·) if the density is wrt the counting measure on a countable subset of R.