Showing papers on "Section (fiber bundle) published in 1999"

A Hitchin-Kobayashi correspondence for Kaehler fibrations

Abstract: Let $X$ be a compact Kaehler manifold and $E\to X$ a principal $K$ bundle, where $K$ is a compact connected Lie group. Let ${\cal A}^{1,1}$ be the set of connections on $E$ whose curvature lies in $\Omega^{1,1}(E\times_{Ad} {\frak k})$, where ${\frak k}$ is the Lie algebra of $K$. Endow $\frak k$ with a nondegenerate biinvariant bilinear pairing. This allows to identify $\{\frak k}\simeq{\frak k}^*$. Let $F$ be a Kaehler left $K$-manifold and suppose that there exists a moment map $\mu$ for the action of $K$ on $F$. Let ${\cal S}=\Gamma(E\times_K F)$. In this paper we study the equation $$\Lambda F_A+\mu(\Phi)=c$$ for $A\in {\cal A}^{1,1}$ and a section $\Phi\in {\cal S}$, where $c\in{\frak k}$ is a fixed central element. We study which orbits of the action of the complex gauge group on ${cal A}^{1,1}\times{\cal S}$ contain solutions of the equation, and we define a positive functional on ${cal A}^{1,1}\times{\cal S}$ which generalises the Yang-Mills-Higgs functional and whose local minima coincide with the solutions of the equation.

Tectonic Evolution of the Careón Ophiolite (Northwest Spain):A Remnant of Oceanic Lithosphere in the Variscan Belt

Abstract: Analysis of the Careon Unit in the Ordenes Complex (northwest Iberian Massif) has supplied relevant data concerning the existence of a Paleozoic oceanic lithosphere, probably related to the Rheic realm, and the early subduction‐related events that were obscured along much of the Variscan belt by subsequent collision tectonics. The ophiolite consists of serpentinized harzburgite and dunite in the lower section and a crustal section made up of coarse‐grained and pegmatitic gabbros. An Early Devonian zircon age ( \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} ewcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} ormalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}...

The Fučík Spectrum

Abstract: In Section 3.5 and again in Section 6.3, we mentioned the situation in which $$f\left( {x,t} \right)/t \to {\alpha _ \pm }\left( x \right){\text{ }}as{\text{ }}t \to \pm \infty .$$ (7.1.1)

On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties

Abstract: Let X be a nonsingular simply connected projective variety of dimension m, E a rank n vector bundle on X, and L a line bundle on X. Suppose that $S^2(E^{*}) \otimes L$ is an ample vector bundle and that there is a constant even rank $r \ge 2$ symmetric bundle map $E \to E^{*} \otimes L$. We prove that $m \le n-r$. We use this result to solve the constant rank problem for symmetric matrices, proving that the maximal dimension of a linear subspace of the space of $m\times m$ symmetric matrices such that each nonzero element has even rank $r \ge 2$ is $m-r+1$. We explain how this result relates to the study of dual varieties in projective geometry and give some applications and examples.

The outer derivation of a complex Poisson manifold

Abstract: We introduce a canonical outer vector field on a Poisson manifold, also due independently to A. Weinstein. We view it as a global section of the sheaf of Poisson vector fields modulo the subsheaf of hamiltonian vector fields. We study this outer derivation mostly in the case of holomorphic Poisson manifolds.

Bernoulli’s Free-boundary Problem

Abstract: As already mentioned in the introduction Bernoulli’s free-boundary problem arises in ideal fluid dynamics, optimal design, electro chemistry, electro statics, and further applications. In the interior Bernoulli problem a connected domain Ω in ℝ n and a constant Q > 0 are given. The task is to find a subset A⊂Ω and a potential u: Ω\A → ℝ such that $$ \begin{gathered} - \Delta u = 0 in \Omega \backslash A, \hfill \\ u = 0 on \partial \Omega , \hfill \\ u = 1 on \partial A, \hfill \\ \frac{{\partial u}} {{\partial v}} = Q on \partial A \hfill \\ \end{gathered} $$ (see Figure 1.1 on page 2). In the exterior Bernoulli problem ∂A is exterior to ∂Ω with u = 1 on ∂Ω, u = 0 on ∂A and Q < 0 (Figure 14.1). The same problem can be posed for the p-Laplacian (Acker and Meyer [3]). We mainly consider the semilinear problem. Typically the interior Bernoulli problem has two solutions, an elliptic one close to the fixed boundary, and a hyperbolic (low energy) solution far from the boundary. Hyperbolic solutions are more delicate for analysis and numerical approximation. Nevertheless there is a second order trial free-boundary method, the implicit Neumann scheme (Section 14.3), with equally good performance for both types of solutions. See our paper with M. Rumpf [56] for the convergence proof. To begin with we describe the applications in physics and industry in detail.

On the fiber product preserving bundle functors

Abstract: We present a complete description of all fiber product preserving bundle functors on the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. This result is based on several general properties of such functors, which are deduced in the first two parts of the paper.

The Gauss map and a noncompact Riemann-Roch formula for constructible sheaves on semiabelian varieties

Abstract: For an irreducible subvariety Z in an algebraic group G we define a nonnegative integer gdeg(Z) as the degree, in a certain sense, of the Gauss map of Z. It can be regarded as a substitution for the intersection index of the conormal bundle to Z with the zero section of T^*G, even though G may be non-compact. For G a semiabelian variety (in particular, an algebraic torus (C^*)^n) we prove a Riemann-Roch-type formula for constructible sheaves on G, which involves our substitutions for the intersection indices. As a corollary, we get that a perverse sheaf on such a G has nonnegative Euler characteristic, generalizing a theorem of Loeser-Sabbah.

Sur la cohomologie d'un fibre tautologique sur le schema de Hilbert d'une surface

Abstract: We compute the cohomology spaces for the tautological bundle tensor the determinant bundle on the punctual Hilbert scheme H of subschemes of length n of a smooth projective surface X. We show that for L and A invertible vector bundles on X, and w the canonical bundle of X, if $w^{-1}\otimes L$, $w^{-1}\otimes A$ and A are ample vector bundles, then the higher cohomology spaces on H of the tautological bundle associated to L tensor the determinant bundle associated to A vanish, and the space of global sections is computed in terms of $H^0(A)$ and $H^0(L\otimes A)$. This result is motivated by the computation of the space of global sections of the determinant bundle on the moduli space of rank 2 semi-stable sheaves on the projective plane, supporting Le Potier's Strange duality conjecture on the projective plane.

On two-dimensional parametric variational problems

Abstract: Let \(\mathcal{F}(X)\) be a two-dimensional parametric variational integral the Lagrangian F(x,z) of which is positive definite and elliptic, and suppose that \(\Gamma\) is a closed rectifiable Jordan curve in \(\mathbb{R}^{3}\). We then prove that there is a conformally parametrized minimizer of \(\mathcal{F}\) in the class \(\mathcal{C}(\Gamma)\) of surfaces \(X \in H^{1,2}(B, \mathbb{R}^{3}) \cap C^{0}(\overline{B}, \mathbb{R}^{3})\) of the type of the disk B which are bounded by \(\Gamma\). An immediate consequence of this theorem is that the Dirichlet integral and the area functional have the same infima, a result whose proof usually requires a Lichtenstein-type mapping theorem or else Morrey's lemma on \(\epsilon\)-conformal mappings. In addition we show that the minimizer of \(\mathcal{F}\) is Holder continuous in B, and even in \(\overline{B}\) if \(\Gamma\) satisfies a chord-arc condition. In Section 1 it is described how our results are related to classical investigations, in particular to the work of Morrey. Without difficulty our approach can be carried over to two-dimensional surfaces of codimension greater than one.

String structure over the Brownian bridge

Abstract: We give the definition of a line bundle over the Brownian bridge by using its section. This allows us to define a Hilbert space of sections of a line bundle over the Brownian bridge associated to the transgression of a representative of an element of H3(M;Z). We consider the case of a string structure over the Brownian bridge: this allows us to define a Hilbert space of spinor fields over the Brownian bridge, when the first Pontryaguin class of the spin bundle over the manifold is equal to 0.

Partial t -Spreads in PG(2t+1,q)

Abstract: This article first of all discusses the problem of the cardinality of maximal partial spreads in PG(3,q), q square, q>4. Let r be an integer such that 2r \leq q+1 and such that every blocking set of PG(2,q) with at most q+r points contains a Baer subplane. If S is a maximal partial spread of PG(3,q) with q^2+1-r lines, then r=s(\sqrt q+1) for an integer s\ge 2 and the set of points of PG(3,q) not covered byS is the disjoint union of s Baer subgeometriesPG(3,\sqrt{q}) . We also discuss maximal partial spreads in PG(3,p^3), p=p_0^h, p_0 prime, p_0 \geq 5, h \geq 1, p eq 5. We show that if p is non-square, then the minimal possible deficiency of such a spread is equal to p^2+p+1, and that if such a maximal partial spread exists, then the set of points of PG(3,p^3) not covered by the lines of the spread is a projected subgeometryPG(5,p) in PG(3,p^3). In PG(3,p^3),p square, for maximal partial spreads of deficiency\delta \leq p^2+p+1 , the combined results from the preceding two cases occur. In the final section, we discuss t-spreads in PG(2t+1,q), q square or q a non-square cube power. In the former case, we show that for small deficiencies \delta, the set of holes is a disjoint union of subgeometries PG(2t+1,\sqrt{q}), which implies that \delta\equiv 0 \pmod{\sqrt{q}+1} and, when (2t+1)(\sqrt{q}-1)

On Spaces with Hereditarily Compact α-Topologies

Abstract: The aim of this paper is to continue the study of semi-compact spaces, i.e. the spaces whose α-topologies are hereditarily compact. The last section deals with semi-closed graphs and semi-convergence in extremally disconnected spaces.

Second Fundamental Form of a Map

Abstract: In this chapter, we introduce the second fundamental form of a map between semi-Riemannian manifolds. The second fundamental form of a map is used largely in the analysis of certain properties of maps between manifolds in the literature. However, in this chapter, we study the second fundamental form of a map only as much as we need to carry out our work on semi-Riemannian maps. We devote the first section to the definition and basic properties of the second fundamental form of a map between semi-Riemannian manifolds. In section 2, we define affine and harmonic maps between semi-Riemannian manifolds and derive certain properties of affine maps. Then in section 3, we define the divergence of a vector field along a map between semi-Riemannian manifolds and generalize the semi-Riemannian divergence theorem to such vector fields. Finally in section 4, we derive the Bochner identity of a map between semi-Riemannian manifolds which will be used in showing a Riemannian map to be affine under certain curvature conditions. Throughout this chapter, and in all following chapters, let M 1 M 2 and M be manifolds with dimensions n 1, n 2 and n, respectively. Also let (M l, g 1), (M 2, g 2 and (M, g) be semi-Riemannian manifolds of indices v l, v 2 and v with Levi-Civita connections \(\mathop abla \limits^1 ,\mathop abla \limits^2 \) and ∇, respectively, unless otherwise stated.

Triades et familles de courbes gauches

Abstract: Author(s): Hartshorne, Robin; Martin-Deschamps, Mireille; Perrin, Daniel | Abstract: Let $A$ be a noetherian ring and $R_A$ be the graded ring $A[X,Y,Z,T]$. In this article we introduce the notion of a triad, which is a generalization to families of curves in ${\bf P}^3_A$ of the notion of Rao module. A triad is a complex of graded $R_A$-modules $(L_1 \to L_0 \to L_{-1})$ with certain finiteness hypotheses on its cohomology modules. A pseudo-isomorphism between two triads is a morphism of complexes which induces an isomorphism on the functors $ h_0 (L\otimes .)$ and a monomorphism on the functors $h_{-1} (L\otimes .)$. One says that two triads are pseudo-isomorphic if they are connected by a chain of pseudo-isomorphisms. We show that to each family of curves is associated a triad, unique up to pseudo-isomorphism, and we show that the map $\{\hbox{families of curves}\}\to \{\hbox{triads}\}$ has almost all the good properties of the map $\{\hbox{curves}\}\to \{\hbox{Rao modules}\}$. In a section of examples, we show how to construct triads and families of curves systematically starting from a graded module and a sub-quotient (that is a submodule of a quotient module), and we apply these results to show the connectedness of $H_{4,0}$.

Wave Number Selection and Large-Scale-Flow Effects due to a Radial Ramp of the Spacing in Rayleigh-Bénard Convection

Abstract: We present experimental results for wave numbers q{sub s} selected in a thin horizontal fluid layer heated from below. The cylindrical sample had an interior section of uniform spacing d=d{sub 0} for radii r r{sub 0} . For Rayleigh numbers R{sub 0}>R{sub c}=1708 in the interior, straight or slightly curved rolls with an average =q(tilde sign){sub c}+{alpha}{epsilon}{sub 0}({epsilon}{sub 0} {identical_to}R{sub 0}/R{sub c}-1) and q(tilde sign){sub c}

More on quotient polytopes

Abstract: In an earlier paper, it was shown that every abstract polytope is a quotient \( {\cal Q} = {\cal M}(W)/N \) of some regular polytope \( {\cal M}(W) \) whose automorphism group is W, by a subgroup N of W. In this paper, attention is focussed on the quotient \( {\cal Q} \), and various important structures relating to polytopes are described in terms of N′, the stabilizer of a flag of the quotient under an action of W (the ‘flag action’). It is pointed out how N′ may be assumed without loss of generality to equal N. The paper also shows what properties of N′ yield polytopes which are regular, section regular, chiral, locally regular, or locally universal. The aim is to make it more practical to study non-regular polytopes in terms of group theory.

The canonical bundle of a hermitian manifold

Abstract: This note contains a simple formula (Proposition 1 in Section 3) for the curvature of the canonical line bundle on a hermitian manifold, using the Levi-Civita connection (instead of the more usual hermitian connection, compatible with the holomorphic structure). As an immediate application of this formula we derive the following: the six-sphere does not admit a complex structure, orthogonal with respect to any metric in a neighborhood of the round one. Moreover, we obtain such a neighborhood in terms of explicit bounds on the eigen-values of the curvature operator. This extends a theorem of LeBrun.

K3-fibered Calabi-Yau threefolds I, the twist map

Abstract: A construction of Calabi-Yaus as quotients of products of lower-dimensional spaces in the context of weighted hypersurfaces is discussed, including desingularisation. The construction leads to Calabi-Yaus which have a fiber structure, in particular one case has K3 surfaces as fibers. These Calabi-Yaus are of some interest in connection with Type II -heterotic string dualities in dimension 4. A section at the end of the paper summarises this for the non-expert mathematician.

An analytic solution to the Busemann-Petty problem on sections of convex bodies

Abstract: We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R^n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n-2)-nd derivative of the parallel section functions. The affirmative answer to the Busemann-Petty problem for n\le 4 and the negative answer for n\ge 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.

On the fiber product preserving bundle functors

Abstract: Ahsmrc~t: We present a complete description of all fiber product preserving bundle functc’rs on the category of tibered manifolds with m-dimensional bases and fiber preservin g maps with local diffeomorphisms as base maps. This result is based on several general properties of such functors, which are dec.uced in the tirst two part\ of the paper. K~~~~nvrt/.r: I3undle functor. Weil bundle, jet bundle, natural transformation. .41S C/C/.\ c(fic~triott: 58AO5, SXA20.

Towards the Standard Model spectrum from elliptic Calabi-Yau

Abstract: We show that it is possible to construct supersymmetric three-generation models of Standard Model gauge group in the framework of non-simply-connected elliptically fibered Calabi-Yau, without section but with a bi-section. The fibrations on a cover Calabi-Yau, where the model has 6 generations of SU(5) and the bundle is given via the spectral cover description, use a different description of the elliptic fibre which leads to more than one global section. We present two examples of a possible cover Calabi-Yau with a free involution: one is a fibre product of rational elliptic surfaces $dP_9$; another example is an elliptic fibration over a Hirzebruch surface. There we give the necessary amount of chiral matter by turning on in the bundles a further parameter, related to singularities of the fibration and the branching of the spectral cover.

Minimal sections of conic bundles

Abstract: Let the threefold X be a general smooth conic bundle over the projective plane P(2), and let (J(X), Theta) be the intermediate jacobian of X. In this paper we prove the existence of two natural families C(+) and C(-) of curves on X, such that the Abel-Jacobi map F sends one of these families onto a copy of the theta divisor (Theta), and the other -- onto the jacobian J(X). The general curve C of any of these two families is a section of the conic bundle projection, and our approach relates such C to a maximal subbundle of a rank 2 vector bundle E(C) on C, or -- to a minimal section of the ruled surface P(E(C)). The families C(+) and C(-) correspond to the two possible types of versal deformations of ruled surfaces over curves of fixed genus g(C). As an application, we find parameterizations of J(X) and (Theta) for certain classes of Fano threefolds, and study the sets Sing(Theta) of the singularities of (Theta).

Ample Vector Bundles of Curve Genus One

Abstract: We investigate the pairs (X,E) consisting of a smooth complex projective variety X of dimension n and an ample vector bundleE of rank n− 1 on X such thatE has a section whose zero locus is a smooth elliptic curve.

A geometric construction of elliptic conic bundles in $P^4$

Abstract: An elliptic conic bundle in P4 is a smooth surface S ⊂ P4 with a map S → C onto an elliptic curve C, whose general fibers are embedded as smooth conic sections in P4. In this note I give a construction of the general elliptic conic bundle in P4 via Cremona transformations. More precisely I construct the general elliptic conic bundle which has a minimal model isomorphic to a smooth quintic elliptic scroll in P4, i.e. a scroll with minimal selfintersection of a section equal to 1. I do not prove that every conic bundle have a minimal model like that, but that the general one does. For any smooth elliptic quintic scroll S5 in P4 we find reducible curves G∪L consisting of a rational normal quartic curve G which meet the scroll in ten points and a line L which is secant to G and a ruling in the scroll. The quadric hypersurfaces passing through G∪L define a Cremona transformation of P4, and the image under this transformation of the elliptic scroll is a smooth conic bundle in P4. Conversely, any elliptic conic bundle in P4 which have a minimal model isomorphic to a scroll S5, contains reducible curves C2 ∪ C3 consisting of a curve C2 of degree 6 and genus 4 and a curve C3 of degree 9 and genus 6 meeting C2 in six points three on each of a pair of skew lines. The curve C2 lies on a quadric surface, while the curve C3 lies on a rational cubic scroll which meet the quadric surface along the two lines. The cubic hypersurfaces through the union of the quadric and the cubic surface defines a Cremona transformation inverse to the one above, and the image of the elliptic conic bundle is a smooth elliptic scroll of degree 5. A conic bundle S is not minimal, and the points blown up on a minimal model S5 to obtain a surface S may not be chosen in general position. The Cremona transformation gives an extrinsic description in the P4 of S5 of how the points chosen as part of the intersection of S5 and a rational normal quartic curve gives rise to a surface S. An intrinsic characterization of the eight points blown up is given in section 3 and compared with the Cremona construction.

Un phénomène de Hartogsdans les variétés projectives

Abstract: In this article, we study univalent open subsets $U\hookrightarrow V$ , $\mathrm{dim} V\geq 2$ , assuming $V\setminus\bar U$ to be pseudoconcave in Andreotti's sense. We prove an Hartogs's Kugelsatz theorem for such open sets: Let U an open subset in V such that $V\setminus\bar U$ is a pseudoconcave domain in the sense of Andreotti. Then U contains a maximal compact hypersurface H. Moreover, any meromorphic section s, of a vector bundle F over V, defined on (a neighborhood of) $\partial\overset{0}{\overline{U}}$ extends on $U\setminus H$ , and, if s is holomorphic then s extends meromorphically to U, with a polar set in H.

Line bundles in supersymmetric coset models

Abstract: The scalars of an N = 1 supersymmetric sigma-model in 4 dimensions parameterize a Kaehler manifold. The transformations of their fermionic superpartners under the isometries are often anomalous. These anomalies can be canceled by introducing additional chiral multiplets with appropriate charges. To obtain the right charges a non-trivial singlet compensating multiplet can be used. However when the topology of the underlying Kaehler manifold is non-trivial, the consistency of this multiplet requires that its charge is quantized. This singlet can be interpreted as a section of a line bundle. We determine the Kaehler potentials corresponding to the minimal non-trivial singlet chiral superfields for any compact Kaehlerian coset space G/H. The quantization condition may be in conflict with the requirement of anomaly cancelation. To illustrate this, we discuss the consistency of anomaly free models based on the coset spaces E_6/SO(10)xU(1) and SU(5)/SU(2)xU(1)xSU(3).

Convergence speed estimates for the norms of the inverses of large truncated Toeplitz matrices

Abstract: Given a continuous function a on the complex unit circle, let T(a) denote the infinite Toeplitz matrix generated by a and let Tn(a) stand for the (n+1)×(n+1) principal section of T(a). We think of T(a) and Tn(a) as operators on l2 spaces. A classical result by Gohberg and Feldman says that if T(a) is invertible, then so is Tn(a) for all sufficiently large n≥n0 and \(\). Only in 1994 did we realize that in fact \(\). In this paper, we provide estimates for the speed with which \(\) converges to \(\). We prove that in the “generic case” the convergence speed can be estimated by the smoothness of a, whereas in some “exceptional cases” (e.g., if T(a) is Hermitian or triangular) it is not the smoothness of $a$ but the orders of certain zeros which determine the convergence speed. Some of the results are extended to operators on lp spaces.

Local geometrised Rankin-Selberg method for GL(n)

Abstract: Following Laumon [10], to a nonramified $\ell$-adic local system $E$ of rank $n$ on a curve $X$ one associates a complex of $\ell$-adic sheaves $_n{\cal K}_E$ on the moduli stack of rank $n$ vector bundles on $X$ with a section, which is cuspidal and satisfies Hecke property for $E$. This is a geometric counterpart of the well-known construction due to Shalika [17] and Piatetski-Shapiro [16]. We express the cohomology of the tensor product $_n{\cal K}_{E_1}\otimes {_n{\cal K}_{E_2}}$ in terms of cohomology of the symmetric powers of $X$. This may be considered as a geometric interpretation of the local part of the classical Rankin-Selberg method for GL(n) in the framework of the geometric Langlands program.