Embedding problems with local conditions

LetY→X be a connectedG-Galois cover of affine varieties in characteristicp, and supposeG=Γ/P for somep-groupP. We show that there is a connected Γ-Galois coverZ→X dominatingY→X, and thatZ→X can be chosen to have prescribed behavior over a given closed subset ofX. There are several versions of this result, depending on whether ramification is permitted, and whether adelic behavior is prescribed. The results are deduced from a general assertion about embedding problems, which is proven for profinite groups. Supported in part by NSF Grant DMS94-00836.     

Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type

LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup _i→∞{sup _y∈K ‖t ⁱx−Tⁱy∼−‖x−y‖}≦0. IfT ^N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tⁱx−Tⁱy‖≦k _i‖x−y∼,x,y∈K, wherek _i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ₀ (X), be less than one. Research supported by National Science Foundation Grant GP 18045.     

Morfismi finiti tra spazi complessi eq-convessità

Riassunto Siaf: Y→X un morfismo finito tra spazi complessi. Dimostro cheY èq-completo oq-convesso seX è rispettivamenteq-completo oq-convesso.

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Summary Letf: Y→X be a finite morphism between complex spaces. We prove thatY isq-complete orq-convex ifX isq-complete orq-convex.

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Lavoro eseguito nell'ambito del GNSAGA del C.N.R.     

Vector bundles as direct images of line bundles

LetX be a smooth irreducible projective variety over an algebraically closed fieldK andE a vector bundle onX. We prove that, if dimX ≥ 1, there exist a smooth irreducible projective varietyZ overK, a surjective separable morphismf:Z →X which is finite outside an algebraic subset of codimension ≥ 3 inX and a line bundleL onX such that the direct image ofL byf is isomorphic toE. WhenX is a curve, we show thatZ, f, L can be so chosen thatf is finite and the canonical mapH ¹(Z, O) →H ¹(X, EndE) is surjective. Dedicated to the memory of Professor K G Ramanathan     

A non-Bernoulli skew product which is loosely Bernoulli

LetT: Y→Y be the Bernoulli two shift with independent generatorQ={Q ₀,Q ₁} and letS: X→X be a measure preserving bijection. If (S, X) is ergodic then the skew product onX×Y defined by {fx339-1} is aK-automorphism. IfŜ is also Bernoulli we sayS is pre-Bernoulli. J. Feldman showed that ifS is pre-Bernoulli thenS must be loosely Bernoulli. We construct an example to show the converse is false, i.e. anS that is loosely Bernoulli but not pre-Bernoulli.     

Differentiable structures on symplectic reductions

LetG be a complex reductive Lie group with maximal compact subgroupK andG×X X a holomorphic action on a Stein manifoldX. LetR _o andR ₁ be two Kempf-Ness sets arising from moment maps induced by strictly plurisubharmonic,K-invariant, proper functions. Then there is a globalK-equivariant diffeomorphism :XX with (R ₀)=R ₁. In particular, the induced differentiable structures on the categorical quotientX G are diffeomorphic. The proof is based on a variant of Moser's method using time-dependent vector fields. An example shows that the differentiable structures can indeed be different, even though they are isomorphic.     

Holomorphic and algebraic vector bundles on 0-convex algebraic surfaces

LetX be a smooth complex algebraic surface such that there is a proper birational morphism/:X → Y withY an affine variety. Let X_hol be the 2-dimensional complex manifold associated toX. Here we give conditions onX which imply that every holomorphic vector bundle onX is algebraizable and it is an extension of line bundles. We also give an approximation theorem of holomorphic vector bundles on X_hol (X normal algebraic surface) by algebraic vector bundles.     

Test for properness using residue calculus

We will consider global problems in the ringK[X ₁, …,X _n] on the polynomials with coefficients in a subfieldK ofC. LetP=(P ₁, …,P _n):K ⁿ →K ⁿ be a polynomial map such that (P ₁,…,P _n) is a quasi-regular sequence generating a proper ideal, the main thing we do is to use the algebraic residues theory (as described in [5]) as a computational tool to give some result to test when a map (P ₁, …,P _n) is a proper map by computing a finite number of residue symbols.     

Polyhedral banach spaces and extensions of compact operators

LetX be a polyhedral Banach space whose dual is anL ₁(μ) space for some measureμ. Then for each Banach spacesY ⊆Z and each compact operatorT: Y →X there exists a norm preserving compact extension Z →X.     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

Some ℵ0-bounded subsets of stone-čech compactifications

We characterize the maximalm-bounded extension of an arbitrary completely regular Hausdorff spaceX. The other principal results are:Theorem. LetX be a locally compact, σ-compact non-compact space with no more than 2^ℵ0 zero-sets. Then assuming the continuum hypothesis,βX − X can be written as the union of 2^2ℵ0 pairwise disjoint, dense ℵ₀-bounded subspaces.Theorem. LetX be a locally compact, σ-compact metric space without isolated points. Then both the set of remote points ofβX and the complement of this set inβX −X are ℵ₀-bounded.     

Holomorphic vector bundles on a neighborhood of non-Stein compact subsets of Stein spaces

LetK be a compact subset of a complex spaceX. Here we give conditions onX andK assuring the existence of a fundamental systemU of open neighborhoods, ofK such that for everyU∈U there is a holomorphic vector bundleE onU which is not holomorphically trivial.

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Sunto SiaX uno spazio complesso eK∩X un compatto. In questo lavoro diamo condizioni suX eK che garantiscono l'esistenza di un sistema fondamentale di intorni apertiU diK inX tali che per ogniU∈U esiste un fibrato olomorfo non-triviale suU.

     

Code length between Markov processes

LetX ₁ andX ₂ be two mixing Markov shifts over finite alphabet. If the entropy ofX ₁ is strictly larger than the entropy ofX ₂, then there exists a finitary homomorphism ϕ:X ₁→X ₂ such that the code length is anL ^p random variable for allp<4/3. In particular, the expected length of the code ϕ is finite. Research supported by KBN grant 2 P03A 039 15 1998–2001.     

A local version of a result of gamlen and gaudet

LetX _D ^p be the span of the Haar function {h _j: J ε D} inL ^p (1 <p < ∞) endowed withL ^p norm. Then for any finite setD, the spacesX _D ^p andl _*D ^p areK _p-isomorphic whereK _p depends onp only.     

A strong law of large numbers for nonexpansive vector-valued stochastic processes

A mapT: X→X on a normed linear space is callednonexpansive if ‖Tx-Ty‖≤‖x-y‖∀x, y∈X. Let (Ω, Σ,P) be a probability space, an increasing chain of σ-fields spanning Σ,X a Banach space, andT: X→X. A sequence (x_n) of strongly -measurable and stronglyP-integrable functions on Ω taking on values inX is called aT-martingale if . LetT: H→H be a nonexpansive mapping on a Hilbert spaceH and let (x_n) be aT-martingale taking on values inH. If then x _n /n converges a.e. LetT: X→X be a nonexpansive mapping on ap-uniformly smooth Banach spaceX, 1<p≤2, and let (x_n) be aT-martingale (taking on values inX). If then there exists a continuous linear functionalf∈X ^* of norm 1 such that If, in addition, the spaceX is strictly convex, x _n /n converges weakly; and if the norm ofX ^* is Fréchet differentiable (away from zero), x _n /n converges strongly. This work was supported by National Science Foundation Grant MCS-82-02093     

On the Cauchy difference on normed spaces

LetX be a real linear normed space, (G, +) be a topological group, andK be a discrete normal subgroup ofG. We prove that if a continuous at a point or measurable (in the sense specified later) functionf:X →G fulfils the condition:f(x +y) -f(x) -f(y) ∈K whenever ‖x‖ = ‖y‖, then, under some additional assumptions onG,K, andX, there esists a continuous additive functionA :X →G such thatf(x) -A(x) ∈K.     

Daniell integral represented by Radon measures

LetL be a sublattice of the space of real continuous functions defined on a Suslin spaceX, such that at no point all the functions inL vanish. Then it is shown that every Daniell integrall μ:L → IR is representable by a Radon measurem onX: μ(ϕ)=∫ϕdm ∀ϕ∈L. The measurem may be uniquely determined by constraining it to be concentrated on a certain type of subset ofX. The relation betweenL ¹(μ) andL ¹(m) is examined in detail.     

On A(X) → TC(X) for some X

Let X be a connected based space and p be a two-regular prime number. If the fundamental group of X has order p, we compute the two-primary homotopy groups of the homotopy fiber of the trace map A(X) → TC(X) relating algebraic K-theory of spaces to topological cyclic homology. The proof uses a theorem of Dundas and an explicit calculation of the cyclotomic trace map K(ℤ[C_p])→ TC(ℤ[C_p]).     

On spaces of p-adic vector measures

LetX be a Hausdorff zero-dimensional topological space,K(X) the algebra of all clopen subsets of X, E a Hausdorff locally convex space over a non-Archimedean valued field and C _b (X) the space of all bounded continuous -valued functions on X. The space M(K(X),E), of all bounded finitely-additive measures m: K(X) → E, is investigated. If we equip C _b (X) with the topologies β _o , β, β _u , τ _b or β _ob , it is shown that, for E (compete, the corresponding spaces of continuous linear operators from C _b (X) to E (are algebraically isomorphic to certain subspaces of M(K(X),E). The text was submitted by the author in English.     

Rational points of bounded height on Del Pezzo surfaces of degree six

LetK be a number field. Denote byV ₃ a split Del Pezzo surface of degree six overK and by ω its canonical divisor. Denote byW ₃ the open complement of the exceptional lines inV ₃. LetN _{W s}(−ω, X) be the number ofK-rational points onW ₃ whose anticanonical heightH _−ω is bounded byX. Manin has conjectured that asymptoticallyN _{W 3}(−ω, X) tends tocX(logX)³, wherec is a constant depending only on the number field and on the normalization of the height. Our goal is to prove the following theorem: For each number fieldK there exists a constantc _K such thatN _{W 3}(−ω, X)≤c_KX(logX)^3+2r , wherer is the rank of the group of units ofO _K. The constantc _K is far from being optimal. However, ifK is a purely imaginary quadratic field, this proves an upper bound with a correct power of logX. The proof of Manin's conjecture for arbitrary number fields and a precise treatment of the constants would require a more sophisticated setting, like the one used by [Peyre] to prove Manin's conjecture and to compute the correct asymptotic constant (in some normalization) in the caseK=ℚ. Up to now the best result for arbitraryK goes back, as far as we know, to [Manin-Tschinkel], who gives an upper boundN _{W 3}(−ω,X)≤cX^l+ε. The author would like to express his gratitude to Daniel Coray and Per Salberger for their generous and indispensable support.