Showing papers in "Crelle's Journal in 1991"

Abelian covers of algebraic varieties

Abstract: More precisely, under our assumptions V χ e G * the eigensheaf of π# Θχ on which the group acts via the character χ is a line b ndle that we denote by L'. On the other hand, to every component of the branch locus D there corresponds naturally its inertia group, which is a cyclic subgroup of G, and a character of the inertia group. We denote by DH φ the union of all the components of the branch locus that have same inertia group H and same character ψ. The line bundles Lx and the divisors DH{I} are then the required building data.

Motivic decomposition of abelian schemes and the Fourier transform.

Abstract: such that the /-adic realization of h (A) is H (A, Qt). Recall that Chow motives are obtained from the category of smooth projective varieties over a field by a construction of Grothendieck using äs intersection theory the Chow ring tensored with Q. See [Ma] and [Mur], 1.1 and l .6, remark 2 for details. By an intricate argument Shermenev also shows how to express h *(A) in terms of h (A). For Chow motives a decomposition äs in (0.1) with /-adic realization äs above is by no means unique. Shermenev's decomposition in particular depends on choices. In this paper we establish a canonicalfunctorial decomposition äs above not only for abelian varieties but also for abelian schemes over a smooth quasiprojective base 5 over a field. In order to formulate such a Statement we extend parts of Manin's paper [Ma] on motives over a field to the case where the base is S. This is straightforward and we obtain a category of relative Chow motives over S.

On contractions of extremal rays of Fano manifolds.

Abstract: A smooth variety X over complex numbers is called Fano if its anticanonical divisor — Kx is ample. The purpose of this note is to collect some results on contractions of extremal rays (Mori theory) which are interesting from the point of describing Fano manifolds of dimension ^ 4. The theory has been successfully used in dimension 3, see [MM], and there is some hope that a similar approach may work somehow in higher dimensions. Furthermore, contractions of extremal rays of Fano manifolds can be treated äs testing grounds for contractions of rays in general. Although contractions of rays of smooth varieties are well understood in dimensions 2 and 3, see [M2], there are still few things known in higher dimensions, see [An], [Bei] and [Ka]. Most of the results of sections l and 2 of the present paper are applicable in a general set-up, i.e. do not need the assumption on ampleness of

Extensions of symplectic groupoids and quantization

Abstract: An important role of Poisson manifolds is äs intermediate objects between ordinary manifolds, with their commutative algebras of functions, and the \"noncommutative spaces\" of quantum mechanics. Up to now, the usual method for passing from Poisson manifolds to noncommutative algebras has been by deformations [BFFLS], [Dr]. The present paper contains the first steps of an effort to quantize Poisson manifolds \"all at once\" by quantizing their symplectic groupoids. Eventually, we hope to apply this method to produce quantum groups from Poisson groups [Dr]. Our program is described more fully in [W6], which can also serve äs a more complete introduction to the present paper.

Spectra of algebras of analytic functions on a Banach spaces.

Abstract: Fix a complex Banach space X, with open unit ball B. We are interested in studying the uniform algebra HTM (B) of bounded analytic functions on B, and its spectrum M = M (B} consisting of the nonzero complex-valued homomorphisms of H °° (B). The restriction of any φ e M to the dual space X * of X yields a linear functional π (φ) on X *, and the projection φ -> π (φ) maps Jl onto the closed unit ball 5** of the bidual ^** of X. If X is one-dimensional, B is the open unit disk in the complex plane, and π is the fibering discussed by K. Hoffman in [Ho]. In this case the projection π is one-to-one over B, and π ~ maps B homeomorphically onto an open subset of Ji. This probably holds whenever X is finite dimensional, and it can be proved at least whenever B is a finite dimensional ball or polydisk. However, when X is infinite dimensional the picture changes completely. It turns out in this case (Theorem 11.1) that the fibers π(ζ) over points of J?** are all quite large.

Reaction-diffusion systems with time delays: monotonicity, invariance, comparison and convergence.

Abstract: In a recent paper, we have obtained differential inequality results, developed comparison theorems and established invariance criteria for mild solutions of abstract functional differential equations in a Banach space. The primary application for these results is to reaction-diffusion systems in which the «reaction» nonlinearity contains time delays. In the present paper, we apply these results to reaction-diffusion systems with time delays of the form ∂u/∂t=DΔu+F(x,u t (x)), x∈Ω, αu+∂u/∂n=0, x∈∂Ω, where Ω is a bounded open set in R n with smooth boundary, ∂Ω, u∈R n , D=diag(d 1 , ..., d n ) with d i ≥0

A factorization theorem for square area-integrable analytic functions.

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Stability of fixed points for order-preserving discrete-time dynamical systems

Abstract: In this paper we prove a number of results on the stability of fixed points and the stabilization of discrete-time order-preserving dynamical Systems on Banach spaces (that is, ones which are order-preserving with respect to an order structure generated by a cone). One of the reasons for studying discrete-time Systems is that they apply to parabolic partial differential equations with time-periodic coefficients. The corresponding results for continuous-time Systems nearly always follow by a limiting argument.

Fundamental representations of Yangians and singularities of R-matrices

Abstract: The construction of Ä-matrices is closely related to the representation theory of quantum groups. In particular, there is an important class of quantum groups, called Yangians, such that to every irreducible finite-dimensional representation Fof a Yangian is associated a solution Rv(u) of the QYBE which is a rational function of u. For every finite-dimensional complex simple Lie algebra g the associated Yangian F(g) is a deformation of the universal enveloping algebra of the Lie algebra g [i] of polynomials in an indeterminate t with values in g. The algebra F(g) is generated by elements x, J(x) for € g, which are deformations of the generators x, xt of g [i]· Moreover, 7(g) contains the universal enveloping algebra t/(g) äs a subalgebra.

A standard basis approach to syzygies of canonical curves.

Abstract: Let C be a smooth projective curve of genus g defined over C and consider the canonical map cpK:C -> P'-* = P(H»(C9a>c)). is an embedding unless C is hyperelliptic. Moreover by a classical result of M. Noether the image of a nonhyperelliptic curve is projectively normal. In this paper we study the syzygies of the canonical curve C £j P'\", i.e. the minimal free resolution 0 «_ Sc «S ^ F, «F2 «... «Fg_2 «0 of the homogeneous coordinate ring Sc = S/IC äs an S = C[x0, . . . , xg _ J-module. It is an easy consequence of Noether's result that ®S(-p-2y»*^ for /> = !,. . . ,#-3 i.e. that Fp is a module generated by elements in degree /? + ! and p + 2, and that /? _ 2 = S(— g — 1). Moreover the resolution is self-dual and satisfies

Real hypersurfaces in quaternionic space forms.

Abstract: Obviously, every totally umbilical hypersurface of a Riemannian manifold is curvature-adapted. In spaces of constant sectional curvature every hypersurface is curvature-adapted. But in other ambient spaces our definition is restrictive. For example, in non-Euclidean spaces of constant holomorphic sectional curvature the curvature-adapted (real) hypersurfaces are exactly the Hopf hypersurfaces (see [3] for the notion of Hopf hypersurfaces). In locally Symmetrie spaces it turns out that for the investigation of the geometry of curvature-adapted hypersurfaces Jacobi field theory may be very useful ( s can be seen in section 5).

Basic sets of Brauer characters of finite groups of Lie type.

Abstract: 1.1. Generalities. Let G be a finite group, t a prime number and 36 a Union of «f-blocks of G. Let us consider the set of ^-modular Brauer characters of ̂ . For the purpose of this paper we assume Brauer characters to be class functions defined on all of G by letting their value be 0 on *f-singular elements. The characteristic function on the /-regul r elements of G is denoted by γ^ throughout this paper. If χ is an ordinary character of G, let # denote the Brauer character χγ#,.

On a problem of Kadison and Singer.

Abstract: In 1959, R. Kadison and I. Singer [10] raised the question whether every pure state (i.e. an extremal element in the space of states) on the C*-algebra D of the diagonal operators on (2 has a unique extension to a (pure) state on B(£2), the C*-algebra of all bounded linear operators on the Hubert space /2. A positive solution to this problem yields clearly a simple characterization of the pure states on B(^2) since on O, which is a commutative algebra, a state is just a probability measure and a pure state a point evaluation on the spectrum of D i. e. on the compactification of the integers.

Critical point theory with symmetries.

Abstract: We also provide means for Computing this category in terms of cohomology. A natural way of obtaining a lower bound for it is by adequately adapting the classical notion of \"cup-length\" to the relative equivariant Situation. Our \"length\" has several advantages. It is defined for every compact Lie group and every equivariant cohomology theory. This makes it very flexible. In practice, the choice of the cohomology theory will depend on the group we are interested in. Here we shall deal with Borel cohomology, and our concrete applications will be for tori S x ... S and /?-tori Z//? ... IIp. Other applications will appear separately. In [BaClPu] stable equivariant cohomotopy will be applied to obtain results for finite groups and some classical groups. In [BaCl] it will be shown that our \"length\" has all nice properties an index theory should have. This is used to obtain bifurcation results.

Solute transport in porous media with equilibrium and non-equilibrium multiple-site adsorption : travelling waves

Abstract: We study travelling wave solutions of a model describing solute transport through porous media, where the chemical species undergoes absorption, i.e. a retention/release reaction with the surface of the porous sceleton. Processes of this type are of fundamental importance in different fields. We do not consider the process at the microscopic scale of single grains and pores, but at the microscopic scale appropriate for the measurement of the phenomena described above

Location, multiplicity and Morse indices of min-max critical points.

Abstract: Our main goal in this paper is, first to relax the boundary condition (FO) (i. e. to allow sometimes sup φ (B) = c) and second to get some more Information about the location of the critical points obtained by such a procedure. We shall then see that, once such a refined version of the min-max principle is proved, it can be used to derive besides existence results in the \"limiting case\" various old and new results concerning the multiplicity and the Morse indices of the critical points.

Normal degenerations of the complex projective plane.

Abstract: Albeit this investigation is interesting for its own sake, there are also applications to other problems in algebraic geometry. For example, Catanese [Ca] uses normal degenerations of P x P in order to study the Zariski closure of some open subsets of moduli spaces of certain surfaces S of general type. We also believe that the study of normal degenerations can be applied to the classification of normal surfaces. We actually give here a result in this direction.

Symbolic algebras of monomial primes.

Abstract: Determination of when symbolic algebras are finitely generated is central to commutative algebra and algebraic geometry. One of the simplest non-trivial classes of examples, which are not divisorial, are the monomial primes P(a, b, c) a k\\x,y, z]. P(a, b, c) is defined to be the kernel of k\\_x,y,z] -> k[f], f(x,y,z) )-^f(t,t,t). Some references where finite generation of the symbolic algebras of these primes has been studied are [E], [Hb], [Hl], [H2], [H-U] and [V].

Rational intersection cohomology of projective toric varieties.

Abstract: Intersection cohomology has been designed to make important topological methods — duality theory and intersection theory — available in the study of Singular spaces. To be able to apply these tools, it is important to know explicitly the intersection cohomology groups for given singular varieties. Torus embeddings provide an interesting class of in general singular varieties, äs they occur in many situations and äs their structure is easily described by means of combinatorial data.

On extensions of number fields obtained by specializing branched coverings.

Abstract: This paper is about the following question: given a branched covering XK -* P|, defined over a number field K, what can one say about the specializations of it to points of PjJ defined over Kl Or, if p is a prime of K and Kp the corresponding local field, then what can one say about the specializations in this case? Serre asked the latter question for rigid branched coverings of P in [S], p. 689-10, and answered it in the case of three branch points when Kp is the real numbers. In [DF], Debes and Fried almost completely answer this question for Kp the real numbers, without any rigidity assumption or restrictions on the number of branch points. This paper is about the case where p is a non-archimedean (i.e., finite) prime of K. Again, there are no rigidity assumptions or restrictions on the number of branch points.

On simultaneous additive equations, ii

Abstract: As in Wooley [22], we say that the equations (1.1) satisfy ihsp-adic solubility condition if they have a non-trivial solution in /?-adic integers (i. e. a solution with not all the xt zero). If the equations satisfy the/j-adic solubility condition for every rational prime/?, then we say that they satisfy the congruence condition. We then define Γ * (k) = Γ * (ki9 . . . , kt) to be the least integer r such that for all s 2> r, and all cy (l :S ι ̂ /, l ^ 7 g s), the equations (1.1) satisfy the congruence condition.

Hardy spaces on affine symmetric spaces

Abstract: In [20] and [26] a construction of Hardy spaces over semi-simple Lie groups G with Hermitian Symmetrie space is given. The purpose of the present work is to generalize this to affine Symmetrie spaces Xof Hermitian type äs introduced in [16] and [17]. It turns out that the function spaces on X thus obtained are precisely the holomorphic discrete series of X constructed in [16] and [17]. In Chapter 3 we give the precise definition of these representations.

Anisotropic motion of a phase interface.

Abstract: 1. The model. In the model we assumed that at time t the solid occupies a region Ω (t) c i? whose boundary dQ(t) is a piecewise smooth (C, say) curve, with a finite number of corners Ρ^(ί)9..., /#(0· We derived two equations for the time evolution of Ω (t) (i.e. of its boundary). The first of these two equations is a relation between the normal velocity of any point Q on the boundary, and the orientation and curvature of the front dQ(t) at this particular point Q, and time /. The other equation arises from the requirement that the capillary force be continuous at the corner points Pt(t),...,PN(t).

On desingularized Horrocks-Mumford quintics.

Abstract: We consider projective small resolutions of such quintics. These are Calabi-Yau threefolds with Euler number equal to zero. There are ihreefamilies, corresponding to the \"natural\" small resolution F, which maps to P1 with Abelian surfaces äs general fibres, its \"opposite\" resolution , when all exceptional curves of V -> Kare flopped, and F, which is a different flop. Using results of Aure [1], which we recall along the way, we compute the cubic form -» 3 on the second cohomology group of each resolution. It turns out to be of different type in the three cases (Propositions 3.1, 4.1 and 5.1).

A geometric characterization of the orbits of s-representations.

Abstract: The orbits of the isotropy representation of semisimple Riemannian Symmetrie spaces, from a geometric point of view, have drawn the attention of mathematicians for many years. The good understanding of these submanifolds turns out to be very important for the understanding of more general classes of submanifolds which have very simple geometric invariants. The purpose of this article is to give a geometric characterization of these orbits (also called orbits of s-representations) in terms of their second fundamental form. Namely, a compact submanifold M of R is an orbit of an s-representation if and only if the tangent bündle TM admits a canonical connection (in a sense that will be defined later) such that the second fundamental form is parallel with respect to this connection in TM and the usual normal connection in the normal bündle. This result provides a generalization of the well known results of Perus [F] about immersions with parallel second fundamental form. In Perus' work the concept of extrinsic Symmetrie submanifold is defined. This has been replaced in our work by the weaker concept of homogeneous submanifold with constant principal curvatures. This means that the holonomy subbundles of the principal normal bündle are extrinsic homogeneous (for a definition see section 1). Note that the last property, if referred to the tangent bündle, characterizes Symmetrie spaces.

Regularity properties of optimal segmentations.

Abstract: Here, R and g correspond to the support and the \"grey levels\" of a given image, while Γ and/correspond to a \"segmentation\" of the image; essentially, Γ is a closed set of piecewise C curves (the \"contours\" of the objects) and/is a regul r approximation of g in each connected component of R — Γ. The minimization of £(/, Γ) thus leads to \"optimal segmentations\" of the image: we refer to [M-S2], Section l, for the motivations leading to such a variational approach and for a beautiful introduction to the entire matter.