Transfer of complex structures and topological invariance of property of being holomorphic

The following question by V. I. Arnold is answered in affirmative. Let X, Y, and Z be three complex manifolds of equal dimension, let p: X → Y be a universal covering, and let g: Y → Z be a nondegenerate holornorphic mapping. Assume that the term Y in the chain X\xrightarrowpY\xrightarrowgZ X\xrightarrow{p}Y\xrightarrow{g}Z is “forgotten,” while the complex structures on X and Z are changed so that the mapping g ∘ p remains holomorphic. Can one recover the “forgotten” term Y? Bibliography: 2 titles.     

Almost Invariant Half-spaces of Algebras of Operators

Given a Banach space X and a bounded linear operator T on X, a subspace Y of X is almost invariant under T if TY í Y+F{TY\subseteq Y+F} for some finite-dimensional “error” F. In this paper, we study subspaces that are almost invariant under every operator in an algebra \mathfrak A{\mathfrak A} of operators acting on X. We show that if \mathfrak A{\mathfrak A} is norm closed then the dimensions of “errors” corresponding to operators in \mathfrak A{\mathfrak A} must be uniformly bounded. Also, if \mathfrak A{\mathfrak A} is generated by a finite number of commuting operators and has an almost invariant half-space (that is, a subspace with both infinite dimension and infinite codimension) then \mathfrak A{\mathfrak A} has an invariant half-space.     

Tableau complexes

Let X, Y be finite sets and T a set of functions X → Y which we will call “ tableaux”. We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such tableau complexes have many nice properties, and are frequently homeomorphic to balls, which we prove using vertex decompositions [BP79]. In our motivating example, the facets are labeled by semistandard Young tableaux, and the more general interior faces are labeled by Buch’s set-valued semistandard tableaux. One vertex decomposition of this “Young tableau complex” parallels Lascoux’s transition formula for vexillary double Grothendieck polynomials [La01, La03]. Consequently, we obtain formulae (both old and new) for these polynomials. In particular, we present a common generalization of the formulae of Wachs [Wa85] and Buch [Bu02], each of which implies the classical tableau formula for Schur polynomials.     

Cantor set and interpolation

In 1998, Y. Benyamini published interesting results concerning interpolation of sequences using continuous functions ℝ → ℝ. In particular, he proved that there exists a continuous function ℝ → ℝ which in some sense “interpolates” all sequences (x _n ) _n∈ℤ ∈ [0, 1]^ℤ “simultaneously.” In 2005, M.R. Naulin and C. Uzcátegui unified and generalized Benyamini’s results. In this paper, the case of topological spaces X and Y with an Abelian group acting on X is considered. A similar problem of “simultaneous interpolation” of all “generalized sequences” using continuous mappings X → Y is posed. Further generalizations of Naulin-Uncátegui theorems, in particular, multidimensional analogues of Benyamini’s results are obtained.     

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”.     

Towards variational analysis in metric spaces: metric regularity and fixed points

The main results of the paper include (a) a theorem containing estimates for the surjection modulus of a “partial composition” of set-valued mappings between metric spaces which contains as a particlar case well-known Milyutin’s theorem about additive perturbation of a mapping into a Banach space by a Lipschitz mapping; (b) a “double fixed point” theorem for a couple of mappings, one from X into Y and another from Y to X which implies a fairly general version of the set-valued contraction mapping principle and also a certain (different) version of the first theorem.     

Metrizability and Coconnectedness

Solving the problem stated in Sichler and Trnková, Topol. Its Appl., 142: 159–179, 2004, we construct metrics μ, ν on a set P such that the spaces X=(P,μ) and Y=(P,ν) have the same monoid of all continuous selfmaps, the space Y is coconnected (in the sense that every continuous map Y×Y→Y depends on at most one coordinate) while X is not. Also, properties of the forgetful functors Metr → Unif → Top are investigated for the “simultaneous variant” of the above problem. Supported by the Grant Agency of Czech Republic under grant 201/06/0664 and by the project of Ministry of Education of Czech Republic MSM 0021620839.     

Obstruction Theory and Coincidences in Positive Codimension

Let f, g ： X→Y be two maps between closed manifolds with dim X ≥ dim Y = n ≥ 3. We study the primary obstruction on（f, g） to deforming f and g to be coincidence free on the n-th skeleton of X. We give examples for which obstructions to deforming f and g to be coincidence free are detected by on （f, g）.     

Banach-stone theorems and separating maps

LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH ⁻¹ are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. Research supported by the Fundació Caixa Castelló, MI/25.043/92     

On log canonical divisors that are log quasi-numerically positive

Let (X Δ) be a four-dimensional log variety that is projective over the field of complex numbers. Assume that (X, Δ) is not Kawamata log terminal (klt) but divisorial log terminal (dlt). First we introduce the notion of “log quasi-numerically positive”, by relaxing that of “numerically positive”. Next we prove that, if the log canonical divisorK _X+Δ is log quasi-numerically positive on (X, Δ) then it is semi-ample.     

On moduli of vector bundles on surfaces with negative Kodaira dimension

Summary Here we prove the following result. Fix integersq, τ,a’, b’, a’ _i, 1≤i≤τ,a’, b’, a’ _i, 1≤i≤τ; then there is an integerew such that for every integert≥w, for every algebraically closed fieldK for every smooth complete surfaceX with negative Kodaira dimension, irregularityq andK _X ² =8(1−q)−τ, the following condition holds; ifX→S is a sequence fo τ blowing-downs which gives a relatively minimal model with ruling ρ:S→C, take as basis of the Neron Severi groupNS(X) a smooth rational curve which is the total transform of a fiber ofC, the total transform of a minimal section of ρ and the total transformD _i, 1≤i≤τ, of the exceptional curver; then for everyH andL∈Pic (X) withH ample,H (resp.L) represented by the integersa’, b’, a’ _i, (resp.a’, b’, a’ _i), 1≤i≤τ, in the chosen basis ofNS(X) the moduli spaceM(ZX, 2,H, L, t) of rank 2H-stable vector bundles onX with determinantL andc ₂=t is generically smooth and the number, dimension and ?birational structure? of the irreducible components ofM(X, 2,H, L, t)_red do not depend on the choice ofK andX. Furthermore the birational structure of these irreducible components can be loosely described in terms of the birational structure of the components of suitableM(S, 2,H’, L’, t’)_red withS a relatively minimal model ofX.

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Sunto SiaX una superficie algebrica liscia completa con dimensione di Kodaira negativa e definita su un campo algebricamente chiusoK; fissiamoH eL∈Pic (X),t∈Z; siaq l’irregolarità diX e τ≔8(1−q)−K _X ^Emphasis>2 ; siaM(X, 2,H, L, t) to schema dei moduli dei fibrati vettorialiH-stabili di rango 2 suX con determinateL ec ₂=t. Si dimostra che esiste una costantew che dipende solo daq, da τ e dalla classe numerica diH e diL (ma non da char (K) o dalla classe di isomorphismo diX) tale che per ognit≥w il numero, la dimensione e ?la struttura birazionale? delle componenti irriducibili diM(X, 2,H, L, t)_red non dipende dalla scelta di char (K),K eX ma solo daq, τ e dalle classi diH eL inNS(X). Inoltre la ?struttura birazionale? di queste componenti irriducibili può essere grossolanamente descritta in termini delle componenti di opportuni spazi di moduliM(S, 2,H’, L’, t’) (doveS è un modello minimale diX).

     

On potentially K 5-H-graphic sequences

Let K _m -H be the graph obtained from K _m by removing the edges set E(H) of H where H is a subgraph of K _m . In this paper, we characterize the potentially K ₅-P ₄ and K ₅-Y ₄-graphic sequences where Y ₄ is a tree on 5 vertices and 3 leaves. Research was supported by NNSF of China (10271105) and by NSF of Fujian (Z0511034), Fujian Provincial Training Foundation for “Bai-Quan-Wan Talents Engineering”, Project of Fujian Education Department, Project of Zhangzhou Teachers College and by NSERC.     

Subadditive Separating Maps

Let X and Y denote compact Hausdorff spaces and let K = R (real numbers) or C(complex numbers). C(X) and C(Y) denote the spaces of K-valued continuous functions on X and Y, respectively. A map H : C(X) C(Y) is separating if fg = 0 implies that HfHg = 0. Results about automatic continuity and the form of additive and linear separating maps have been developed in [1], [2], [3], [4], [5], [7], [8], and [10]. In this article similar results are developed for subadditive separating maps. We show (Theorem 5.11) that certain biseparating, subadditive bijections H are automatically continuous.     

Relative Gromov-Witten invariants and the mirror formula

Let X be a smooth complex projective variety, and let be a smooth very ample hypersurface such that is nef. Using the technique of relative Gromov-Witten invariants, we give a new short and geometric proof of (a version of) the “mirror formula”, i.e. we show that the generating function of the genus zero 1-point Gromov-Witten invariants of Y can be obtained from that of X by a certain change of variables (the so-called “mirror transformation”). Moreover, we use the same techniques to give a similar expression for the (virtual) numbers of degree-d plane rational curves meeting a smooth cubic at one point with multiplicity 3d, which play a role in local mirror symmetry. Received: 11 July 2001 / Published online: 4 February 2003 Funded by the DFG scholarships Ga 636/1–1 and Ga 636/1–2.     

The Abelian/Nonabelian correspondence and Frobenius manifolds

We propose an approach via Frobenius manifolds to the study (began in [BCK2] of the relation between rational Gromov–Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov–Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.     

Anticanonical models of ruled surfaces

Summary One investigates the properties of the so-called anticanonical model of a ruled non-rational surfaceX with the anticanonical dimension 2 (see theorem 2 below). In such a way one gets a ?non-rational? analogue of a result of F. Sakai (see [12], and also theorem 1 below). As an application of these two theorems one characterizes the normal projective surfacesY overC with the property that there is a positive integerr such that—rK _Y is an ample Cartier divisor, whereK _y is a canonical (Weil) divisor ofY (see corollary 8 below).

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Riassunto Si studiano i modelli anticanonici delle superficie rigate non-razionali di dimensione anticanonica 2 (teorema 2). Si trova un risultato simile a quello di F. Sakai (teorema 1). Come applicazione si dà una caratterizzazione delle superficie proiettive normaliY sul campoC tali che un multiplo—rK _Y, conr>0, è un divisore di Cartier ampio, doveK _y è un divisore (di Weil) canonico (Corollario 8).

     

The distinguishing number of the direct product and wreath product action

Let G be a group acting faithfully on a set X. The distinguishing number of the action of G on X, denoted D _G(X), is the smallest number of colors such that there exists a coloring of X where no nontrivial group element induces a color-preserving permutation of X. In this paper, we consider the distinguishing number of two important product actions, the wreath product and the direct product. Given groups G and H acting on sets X and Y respectively, we characterize the distinguishing number of the wreath product G ≀_Y H in terms of the number of distinguishing colorings of X with respect to G and the distinguishing number of the action of H on Y. We also prove a recursive formula for the distinguishing number of the action of the Cartesian product of two symmetric groups S _m × S _n on [m] × [n].     

Ample subvarieties and rationally connected fibrations

Under some positivity assumptions, extension properties of rationally connected fibrations from a submanifold to its ambient variety are studied. Given a family of rational curves on a complex projective manifold X inducing a covering family on a submanifold Y with ample normal bundle in X, the main results relate, under suitable conditions, the associated rational connected fiber structures on X and on Y. Applications of these results include an extension theorem for Mori contractions of fiber type and a classification theorem in the case Y has a structure of projective bundle or quadric fibration. All authors acknowledge support by MIUR National Research Project “Geometry on Algebraic Varieties” (Cofin 2004). The research of the second author was partially supported by NSF grants DMS 0111298 and DMS 0548325. The third author acknowledges partial support by the University of Milan (FIRST 2003).     

Patterson-Sullivan Theory in Higher Rank Symmetric Spaces

Let X = G/K be a Riemannian symmetric space of noncompact type and a discrete “generic” subgroup of G with critical exponent . Denote by the set of regular elements of the geometric boundary of X. We show that the support of all -invariant conformal densities of dimension on (e.g. Patterson-Sullivan densities) lie in a same and single regular G-orbit on . This provides information on the large-scale growth of -orbits in X. If in addition we assume to be of divergence type, then there is a unique density of the previous type. Furthermore, we explicitly determine and this G-orbit for lattices, and show that they are of divergence type. Submitted: November 1997, revised: January 1999.