Some remarks on the study of good contractions

LetX be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model Theory, i.e. a projective morphism φ:X→Z onto a normal varietyZ with connected fibers which is given by a (high multiple of a) divisor of the typeK _x+rL, wherer is a positive rational number andL is an ample Cartier divisor. We first prove that the dimension of anu fiberF of φ is bigger or equal to (r-1) and, if φ is birational, thatdimF≥r, with the equalities if and only ifF is the projective space andL the hyperplane bundle (this is a sort of “relative” version of a theorem of Kobayashi-Ochiai). Then we describe the structure of the morphism φ itself in the case in which all fibers have minimal dimension with the respect tor. If φ is a birational divisorial contraction andX has terminal singularities we prove that φ is actually a “blow-up”.     

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.     

On the comparison theorem for étale cohomology of non-Archimedean analytic spaces

Let ϕ:Y →X be a morphism of finite type between schemes of locally finite type over a non-Archimedean fieldk, and letF be an étale constructible sheaf onY. In [Ber2] we proved that if the torsion orders ofF are prime to the characteristic of the residue field ofk then the canonical homomorphisms (R ^Q ϱ*F)^an →R ^q ϱ _* ^an F ^an are isomorphisms. In this paper we extend the above result to the class of sheavesF with torsion orders prime to the characteristic ofk.     

Classification of Homomorphisms from C (X) to Simple C*-Algebras of Real Rank Zero

Abstract Let A be a unital simple C*-algebra of real zero, stable rank one, with weakly unperforated K ₀₍ A) and unique normalized quasi-trace τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φ and ψ induce the same element in KL(C(X), A) and the two lineal functionals τ∘φ and τ∘ψ are equal. We also show that, with an injectivity condition, an almost multiplicative morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism. Research partially supported by NSF Grants DMS 93-01082 (H.L) and DMS-9401515(G.G). This work was reported by the first named author at West Coast Operator Algebras Seminar (Sept. 1995, Eugene, Oregon)     

A new conservation result of WKL0 over RCA0

In this paper we give a partial answer to a conjecture of Tanaka. We prove that: if WKL₀ proves a sentence of the form (∀X)(∃!Y)ψ(X, Y) for a Σ⁰ ₃-formula ψ, then so does RCA₀. Received: 12 April 1999 / Published online: 3 October 2001     

Uniformity over primes of tamely ramified splittings

We fix a prime p and let f(X) vary over all monic integer polynomials of fixed degree n. Given any possible shape of a tamely ramified splitting of p in an extension of degree n, we prove that there exists a rational function φ(X)∈ℚ(X) such that the density of the monic integer polynomials f(X) for which the splitting of p has the given shape in ℚ[X]/f(X) is φ(p) (here reducible polynomials can be neglected). As a corollary, we prove that, for p≥n, the density of irreducible monic polynomials of degree n in ℤ _p [X] is the value at p of a rational function φ _n (X)∈ℚ(X). All rational functions involved are effectively computable. Received: 15 September 1998 / Revised version: 21 October 1999     

On States of Total Weighted Occupation Times for Superdiffusions

Suppose X is a superdiffusion in R ^d with general branching mechanism ψ, and Y _{r D} denotes the total weighted occupation time of X in a bounded smooth domain D. We discuss the conditions on ψ to guarantee that Y _{r D} has absolutely continuous states. And for particular ψ₍ z) = z ^1+β, 0 < β≤ 1, we prove that, in the case d < 2 + 2/β, Y _{r D} is absolutely continuous with respect to the Lebesgue measure in , whereas in the case d > 2 + 2/β, it is singular. As we know the absolute continuity and singularity of Y _{r D} have not been discussed before. Received February 1, 1999, Revised February 25, 2000, Accepted March 9, 2000     

Local properties of the solution set of the operator equation in Banach spaces in a neighbourhood of a bifurcation point

In this work we study the problem of the existence of bifurcation in the solution set of the equation F(x, λ)=0, where F: X×R ^k →Y is a C ²-smooth operator, X and Y are Banach spaces such that X⊂Y. Moreover, there is given a scalar product 〈·,·〉: Y×Y→R ¹ that is continuous with respect to the norms in X and Y. We show that under some conditions there is bifurcation at a point (0, λ₀)∈X×R ^k and we describe the solution set of the studied equation in a small neighbourhood of this point.     

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     

The range of a projection of small norm inl 1 n

It is proved that there exists a positive function Φ(∈) defined for sufficiently small ∈ 〉 0 and satisfying lim_t→0 Φ(∈) =0 such that for any integersn 〉>0, ifQ is a projection ofl ₁ ⁿ onto ak-dimensional subspaceE with ‖|Q‖|≦1+∈ then there is an integerh〉=k(1−Φ(∈)) and anh-dimensional subspaceF ofE withd(F,l ₁ ^h ) 〈= 1+Φ (∈) whered(X, Y) denotes the Banach-Mazur distance between the Banach spacesX andY. Moreover, there is a projectionP ofl ₁ ⁿ ontoF with ‖|P‖| ≦1+Φ(∈). Author was partially supported by the N.S.F. Grant MCS 79-03042.     

On linear selections of convex set-valued maps

We study continuous subadditive set-valued maps taking points of a linear space X to convex compact subsets of a linear space Y. The subadditivity means that φ(x ₁ + x ₂) ⊂ φ(x ₁) + φ(x ₂). We characterize all pairs of locally convex spaces (X, Y) for which any such map has a linear selection, i.e., there exists a linear operator A: X → Y such that Ax ∈ φ(x), x ∈ X. The existence of linear selections for a class of subadditive maps generated by differences of a continuous function is proved. This result is applied to the Lipschitz stability problem for linear operators in Banach spaces.     

Weak local homeomorphisms and <Emphasis Type="Italic">B</Emphasis>-favorable spaces

Let X and Y be topological spaces such that an arbitrary mapping f: X → Y for which every preimage f ⁻¹ (G) of a set G open in Y is an F _σ-set in X can be represented in the form of the pointwise limit of continuous mappings f _n : X → Y. We study the problem of subspaces Z of the space Y for which the mappings f: X → Z possess the same property. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1189–1195, September, 2008.     

Daugavet centers and direct sums of Banach spaces

A linear continuous nonzero operator G: X → Y is a Daugavet center if every rank-1 operator T: X → Y satisfies ||G + T|| = ||G|| + ||T||. We study the case when either X or Y is a sum X _1⊕F X ₂ of two Banach spaces X ₁ and X ₂ by some two-dimensional Banach space F. We completely describe the class of those F such that for some spaces X ₁ and X ₂ there exists a Daugavet center acting from X _1⊕F X ₂, and the class of those F such that for some pair of spaces X ₁ and X ₂ there is a Daugavet center acting into X _1⊕F X ₂. We also present several examples of such Daugavet centers.     

On Chow and Brauer groups of a product of Mumford curves

Let C₁,···,C_d be Mumford curves defined over a finite extension of and let X=C₁×···×C_d. We shall show the following: (1) The cycle map CH₀(X)/n → H^2d(X, μ_n^⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH₀(X)→Hom(Br(X),) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH₀(X).     

Estimates of the weak distance between finite-dimensional Banach spaces

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫_Ω S(ω)oT(ω)dω is the identity operator inX and ∫_Ω||S(ω)||·||T(ω)||dω=O(n ^α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube. Research supported in part by a grant of the Israel Academy of Sciences.     

Construction of orthonormal multi-wavelets with additional vanishing moments

An iterative scheme for constructing compactly supported orthonormal (o.n.) multi-wavelets with vanishing moments of arbitrarily high order is established. Precisely, let φ=[φ₁,. . .,φ_r]^⊤ be an r-dimensional o.n. scaling function vector with polynomial preservation of order (p.p.o.) m, and ψ=[ψ₁,. . .,ψ_r]^⊤ an o.n. multi-wavelet corresponding to φ, with two-scale symbols P and Q, respectively. Then a new (r+1)-dimensional o.n. scaling function vector φ^♯:=[φ^⊤,φ_r+1]^⊤ and some corresponding o.n. multi-wavelet ψ^♯ are constructed in such a way that φ^♯ has p.p.o.=n>m and their two-scale symbols P^♯ and Q^♯ are lower and upper triangular block matrices, respectively, without increasing the size of the supports. For instance, for r=1, if we consider the mth order Daubechies o.n. scaling function φ_m^D, then φ^♯:=[φ_m^D,φ₂]^⊤ is a scaling function vector with p.p.o. >m. As another example, for r=2, if we use the symmetric o.n. scaling function vector φ in our earlier work, then we obtain a new pair of scaling function vector φ^♯=[φ^⊤,φ₃]^⊤ and multi-wavelet ψ^♯ that not only increase the order of vanishing moments but also preserve symmetry. Dedicated to Charles A. Micchelli in Honor of His Sixtieth Birthday Mathematics subject classifications (2000) 42C15, 42C40. Charles K. Chui: Supported in part by NSF grants CCR-9988289 and CCR-0098331 and Army Research Office under grant DAAD 19-00-1-0512. Jian-ao Lian: Supported in part by Army Research Office under grant DAAD 19-01-1-0739.     

Direct images of bundles under Frobenius morphism

Let X be a smooth projective variety of dimension n over an algebraically closed field k with char(k)=p>0 and F:X→X ₁ be the relative Frobenius morphism. For any vector bundle W on X, we prove that instability of F _* W is bounded by instability of W⊗T^ℓ(Ω¹ _X ) (0≤ℓ≤n(p-1)) (Corollary 4.9). When X is a smooth projective curve of genus g≥2, it implies F _* W being stable whenever W is stable. Dedicated to Professor Zhexian Wan on the occasion of his 80th birthday.     

The Subadditive Ergodic Theorem and generic stretching factors for free group automorphisms

Let F _k be a free group of rank k ≥ 2 with a fixed set of free generators. We associate to any homomorphism φ from F _k to a group G with a left-invariant semi-norm a generic stretching factor, λ(φ), which is a noncommutative generalization of the translation number. We concentrate on the situation where φ: F _k → Aut(X) corresponds to a free action of F _k on a simplicial tree X, in particular, where φ corresponds to the action of F _k on its Cayley graph via an automorphism of F _k . In this case we are able to obtain some detailed “arithmetic” information about the possible values of λ = λ(φ). We show that λ ≥ 1 and is a rational number with 2kλ ∈ ℤ[1/(2k − 1)] for every φ ∈ Aut(F _k ). We also prove that the set of all λ(φ), where φ varies over Aut(F _k ), has a gap between 1 and 1+(2k−3)/(2k ²−k), and the value 1 is attained only for “trivial” reasons. Furthermore, there is an algorithm which, when given φ, calculates λ(φ). The second and the third author were supported by the NSF grant DMS#0404991 and the NSA grant DMA#H98230-04-1-0115.