Showing papers in "Crelle's Journal in 1998"
TL;DR: Brou et al. as mentioned in this paper introduced monodromy representations of the braid groups which factorize through the Hecke algebras extending results of Cherednik Opdam Kohno and others.
Abstract: Presentations a la Coxeter are given for all irreducible nite com plex re ection groups They provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite Coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra Hecke algebras We introduce monodromy representations of the braid groups which factorize through the Hecke algebras extending results of Cherednik Opdam Kohno and others Summary Introduction Complex re ection groups and their presentations A Background from complex re ection groups B Presentations Braid groups and their diagrams A Generalities about hyperplane complements B Generalities about the braid groups C The braid diagrams Proofs of the main theorems for the braid groups B de e r A Notation and prerequisites B Computation of B de e r and of its center for d C Computation of B e e r and of its center Hecke algebras A Background from di erential equations and monodromy B A family of monodromy representations of the braid group C Hecke algebras Diagrams and tables Appendix Generators of the monodromy around a divisor Appendix tables to Mathematics Subject Classi cation Primary G We thank Jean Michel Peter Orlik Pierre Vogel for useful conversations and the Isaac Newton Institute for its hospitality while the last version of this manuscript was written up The second named author gratefully acknowledges nancial support by the Fondation Alexander von Humboldt for his stays in Paris Michel Brou e Gunter Malle Raphael Rouquier
367 citations
TL;DR: In this article, a canonical free resolution for arbitrary monomial modules and lattice ideals was constructed for the class of ideals defining toric varieties, including monomial ideals and defining ideals of toric variety.
Abstract: We construct a canonical free resolution for arbitrary monomial modules and lattice ideals. This includes monomial ideals and defining ideals of toric varieties, and it generalizes our joint results with Irena Peeva for generic ideals [BPS],[PS]. Introduction Given a field k, we consider the Laurent polynomial ring T = k[x±1 1 , . . . , x ±1 n ] as a module over the polynomial ring S = k[x1, . . . , xn]. The module structure comes from the natural inclusion of semigroup algebras S = k[N] ⊂ k[Z] = T . A monomial module is an S-submodule of T which is generated by monomials x = x1 1 · · ·xan n , a ∈ Z. Of special interest are the two cases when M has a minimal monomial generating set which is either finite or forms a group under multiplication. In the first case M is isomorphic to a monomial ideal in S. In the second case M coincides with the lattice module ML := S {x | a ∈ L} = k {x | b ∈ N + L} ⊂ T. for some sublattice L ⊂ Z whose intersection with N is the origin 0 = (0, . . . , 0). We shall derive free resolutions of M from regular cell complexes whose vertices are the generators of M and whose faces are labeled by the least common multiples of their vertices. The basic theory of such cellular resolutions is developed in Section 1. Our main result is the construction of the hull resolution in Section 2. We rescale the exponents of the monomials in M , so that their convex hull in R is a polyhedron Pt whose bounded faces support a free resolution of M . This resolution is new and interesting even for monomial ideals. It need not be minimal, but, unlike minimal resolutions, it respects symmetry and is free from arbitrary choices. In Section 3 we relate the lattice module ML to the Z/L-graded lattice ideal IL = 〈 x − x | a− b ∈ L 〉 ⊂ S. This class of ideals includes ideals defining toric varieties. We express the cyclic Smodule S/IL as the quotient of the infinitely generated S-module ML by the action of L. In fact, we like to think of ML as the “universal cover” of IL. Many questions about IL can thus be reduced to questions about ML. In particular, we obtain the hull resolution of a lattice ideal IL by taking the hull resolution of ML modulo L. This paper is inspired by the work of Barany, Howe and Scarf [BHS] who introduced the polyhedron Pt in the context of integer programming. The hull resolution generalizes results in [BPS] for generic monomial ideals and in [PS] for generic lattice ideals. In these generic cases the hull resolution is minimal.
220 citations
TL;DR: In this article, a large number of moduli spaces of stable bun-dles on a general algebraic elliptic surface were computed using a new class of relative Fourier-Mukai transforms.
Abstract: We compute a large number of moduli spaces of stable bun- dles on a general algebraic elliptic surface using a new class of relative Fourier-Mukai transforms.
151 citations
TL;DR: In this article, it was shown that the algebraic monodromy groups associated to a motive over a number field are generated by certain one-parameter subgroups determined by Hodge numbers.
Abstract: We prove that the `-adic algebraic monodromy groups associated to a motive over a number field are generated by certain one-parameter subgroups determined by Hodge numbers. In the special case of an abelian variety we obtain stronger statements saying roughly that the `-adic algebraic monodromy groups look like a Mumford-Tate group of some (other?) abelian variety. When the endomorphism ring is Z and the dimension satisfies certain numerical conditions, we deduce the Mumford-Tate conjecture for this abelian variety. We also discuss the problem of finding places of ordinary reduction.
127 citations
TL;DR: An approach is proposed based on an analysis of the corresponding linear system on a degeneration of the plane itself, leading to a simple recursion for these dimensions, obtaining results in the ``quasi-homogeneous'' case when all the multiplicities are equal except one.
Abstract: Fixing n general points pi in the plane, what is the dimension of the space of plane curves of degree d having multiplicity mi at pi for each i? In this article we propose an approach to attack this problem, and demonstrate it by successfully computing this dimension for all n and for mi constant, at most 3. This application, while previously known (see (Hi1)), demonstrates the utility of our approach, which is based on an analysis of the corresponding linear system on a degeneration of the plane itself, leading to a simple recursion for these dimensions. We also obtain results in the "quasi-homogeneous" case when all the multiplicities are equal except one; this is the natural family to consider in the recursion.
120 citations
118 citations
TL;DR: In this paper, it was shown that the Hardy-Littlewood maximal operator is bounded in the Sobolev space, and that the maximal function often has partial derivatives, thus preserving the zero boundary values.
Abstract: In this note we show that the local Hardy-Littlewood maximal operator is bounded in the Sobolev space. Thus the maximal function often has partial derivatives. We also show that the maximal operator preserves the zero boundary values in Sobolev’s sense.
114 citations
TL;DR: In this article, a family of finite-dimensional modules with a semisimple action of the subalgebra $A(gl_N)$ is studied and a characterization of irreducible tame modules in terms of their Drinfeld polynomials is given.
Abstract: We study certain family of finite-dimensional modules over the Yangian $Y(gl_N)$. The algebra $Y(gl_N)$ comes equipped with a distinguished maximal commutative subalgebra $A(gl_n)$ generated by the centres of all algebras in the chain $Y(gl_1)\subset Y(gl_2)\subset...\subset Y(gl_N)$. We study the finite-dimensional $Y(gl_N)$-modules with a semisimple action of the subalgebra $A(gl_N)$. We call these modules tame.
We provide a characterization of irreducible tame modules in terms of their Drinfeld polynomials. We prove that every irreducible tame module splits into a tensor product of modules corresponding to the skew Young diagrams and some one-dimensional module.
The eigenbases of $A(gl_N)$ in irreducible tame modules are called Gelfand-Zetlin bases. We provide explicit formulas for the action of the Drinfeld generators of the algebra $Y(gl_N)$ on the vectors of Gelfand-Zetlin bases.
113 citations
112 citations
TL;DR: In this article, it was shown that if a and b are distinct nonzero integers, then the Diophantine equations x − az = 1, y − bz = 1 possess at most three solutions in positive integers (x, y, z).
Abstract: It is proven that if a and b are distinct nonzero integers then the simultaneous Diophantine equations x − az = 1, y − bz = 1 possess at most three solutions in positive integers (x, y, z). Since there exist infinite families of pairs (a, b) for which the above equations have at least two solutions, this result is not too far from the truth. If, further, u and v are nonzero integers with av − bu nonzero, then the more general equations x − az = u, y − bz = v are shown to have 2min{ω(u),ω(v)} log (|u|+ |v|) solutions in integers, where ω(m) denotes the number of distinct prime factors of m and the implied constant is absolute. These results follow from a combination of techniques including simultaneous Padé approximation to binomial functions, the theory of linear forms in two logarithms and some gap principles, both new and familiar. Some connections to elliptic curves and related problems are briefly discussed.
110 citations
TL;DR: In this article, an irreducible algebraic curve in the affine complex plane is considered, and it is shown that there are only finitely many points P of S such that both coordinates of P are singular moduli (i.e. invariants of elliptic curves with complex multiplication).
Abstract: Abstract Let S be an irreducible algebraic curve in the affine complex plane. Assume that S is neither a horizontal Une, nor a vertical line, nor a modular curve Y0(N) (for any integer N ≧ 1). Then there are only finitely many points P of S such that both coordinates of P are singular moduli (i.e. invariants of elliptic curves with complex multiplication).
TL;DR: In this paper, the authors develop an approach to noncommutative algebraic geometry in the perturbative regime around ordinary commutative geometry, called NC-schemes.
Abstract: We develop an approach to noncommutative algebraic geometry ``in the perturbative regime" around ordinary commutative geometry. Let R be a noncommutative algebra and A=R/[R,R] its commutativization. We describe what should be the formal neighborhood of M=Spec(A) in the (nonexistent) space Spec(R). This is a ringed space (M,O) where O is a certain sheaf of noncommutative rings on M. Such ringed spaces can be glued together to form more global objects called NC-schemes. We are especially interested in NC-manifolds, NC-schemes for which the completion of O at every point of M is isomorphic to the algebra of noncommutative power series (completion of the free associative algebra). An explicit description of the simplest NC-manifold, the affine space, is given by using the Feynman-Maslov calculus of ordered operators. We show that many familiar algebraic varieties can be naturally enlarged to NC-manifolds. Among these are all the classical flag varieties and all the smooth moduli spaces of vector bundles.
TL;DR: In this article, it was shown that if X is a manifold in P, then the Poisson functional, the Riesz functional, and the Lelong functional associated to a generic α have plurisubharmonic envelopes.
Abstract: Let X be a complex manifold and AX be the family of maps D → X which are holomorphic in a neighbourhood of the closure of the unit disc D. Such maps are called (closed) analytic discs in X. A disc functional on X is a map H : AX → R ∪ {−∞}. The envelope of H is the function EH : X → R ∪ {−∞}, x 7→ inf {H(f) ; f ∈ AX , f(0) = x}. Through work of Evgeny Poletsky, it has transpired that certain disc functionals on domains in C have plurisubharmonic envelopes. There are essentially only three known classes of disc functionals with plurisubharmonic envelopes. The Poisson functional associated to an upper semi-continuous function φ : X → R ∪ {−∞} takes f ∈ AX to 1 2π ∫ T φ ◦ f dλ, where λ is the arc length measure on the unit circle T. The Riesz functional associated to a plurisubharmonic function v on X takes f to 1 2π ∫ D log | · |∆(v ◦ f), where ∆(v ◦ f) is considered as a positive Borel measure on D, equal to zero if v ◦ f = −∞. The Lelong functional associated to a non-negative function α on X takes f to ∑ z∈D α(f(z)) mz(f) log |z|, where mz(f) denotes the multiplicity of f at z. Define P as the class of complex manifolds X for which there exists a finite sequence of complex manifolds and holomorphic maps X0 h1 −−→ . . . hm −−→ Xm = X, m ≥ 0, where X0 is a domain in a Stein manifold and each hi is either a covering (unbranched and possibly infinite) or a finite branched covering (a proper holomorphic surjection with finite fibres). The class P is closed under taking products and passing to subdomains. Besides domains in Stein manifolds, P contains for instance all Riemann surfaces and all covering spaces of projective manifolds. The main result of the paper is that if X is a manifold in P, then the Poisson functional, the Riesz functional associated to a continuous v, and the Lelong functional associated to a generic α have plurisubharmonic envelopes. In each case, the envelope is the supremum of a naturally defined class of plurisubharmonic functions. 2000 Mathematics Subject Classification. Primary: 32F05; secondary: 32C10, 14E20. The first-named author was supported in part by the Natural Sciences and Engineering Research Council of Canada, and by a VPRSC grant from the University of Western Ontario. Typeset by AMS-TEX 1
TL;DR: In this article, a non-integrable or at-MM obius structure on a conformal 2-manifold induces a complex projective structure: the manifold possesses an atlas whose transition functions are complex MM obius transformations.
Abstract: Conformal 2-manifolds possess a fascinatingly rich and elegant theory which can be viewed in many ways: it is the theory of Riemann surfaces in complex analysis, or of complex curves in algebraic geometry. In this paper, a purely diierential geometric point of view will be taken, the aim being to introduce two geometric structures that a conformal 2-manifold might be equipped with, and to study the relationship between them. These structures are closely related to the projective and aane structures of Riemann surface theory. The rst structure can be viewed as a nonintegrable or nonholomorphic version of a complex projective structure, and will be called a MM obius structure. An integrable or at MM obius structure on a conformal 2-manifold induces a complex projective structure: the manifold possesses an atlas whose transition functions are complex MM obius transformations. However, contrary to common usage 9], the MM obius structures discussed herein are not necessarily integrable: they possess a curvature, analogous to the Cotton-York tensor of a conformal 3-manifold, whose vanishing is equivalent to integrability. MM obius structures are also diierent from real projective structures, in much the same way as conformal and real projective structures diier in higher dimensions. (In one dimension, MM obius and real projective structures do coincide and are always integrable.) The other topic of interest here is Einstein-Weyl geometry 3, 8, 14]. This is the geometry of a conformal manifold equipped with a compatible (or conformal) torsion free connection, such that the symmetric tracefree part of the Ricci tensor of this connection vanishes. These manifolds generalise Einstein manifolds in a natural way, and have been investigated in some detail recently (see 2, 6, 8] and references therein). In 11], Ped-ersen and Tod posed the problem of classifying compact two dimensional Einstein-Weyl manifolds|the possible geometries of compact three dimensional Einstein-Weyl mani-folds have been classiied (locally) by Tod 13]. However, the deenition just given of an Einstein-Weyl manifold is vacuous in the two dimensional case and Pedersen and Tod did not ooer an alternative deenition. One of the main goals of this paper is to give explicitly such a deenition and present a classiication of the compact orientable examples.
TL;DR: In this article, a canonical compactification of Hitchin's moduli space of stable Higgs bundles with fixed determinant of odd degree over a Riemann surface, producing a projective variety by gluing in a divisor at infinity, is considered.
Abstract: In this paper we consider a canonical compactification of Hitchin's moduli space of stable Higgs bundles with fixed determinant of odd degree over a Riemann surface, producing a projective variety by gluing in a divisor at infinity. We give a detailed study of the compactified space, the divisor at infinity and the moduli space itself. In doing so we reprove some assertions of Laumon and Thaddeus on the nilpotent cone.
TL;DR: In this paper, the problem of classifying non-trivial one-parameter families of isometries of F = F 0 preserving both principal curvatures was studied.
Abstract: Introduction Let F be a surface in Euclidean 3-space without umbilic points. This paper studies the following Problem : To classify non-trivial one-parameter families F , 2 (?;) of isometries of F = F 0 preserving both principal curvatures. Since the Gaussian curvature is preserved by isometries one can reformulate the problem replacing \"both principle curvatures\" by \"the mean curvature function\". Let us specify what do we mean by a non-trivial family. We consider families of surfaces which do not diier by rigid motions. We suppose also that the surfaces and isometries are suucient smooth. The case of surfaces with constant mean curvature (CMC-surfaces), which all possess non-trivial isometries, is also excluded from our consideration. We suppose that the mean curvature is a non-trivial function on F. It turns out that the condition of possessing a one-parameter family F of isometries, preserving H, implies restrictive conditions on F. Moreover, all the family F can be described (see section 2) as a reparametrization of F itself. The problem is reduced to the problem of classiication of surfaces F. Since the problem formulated at the beginning of this introduction was rst studied by Bonnet, we call these surfaces Bonnet surfaces. The problem is classical and many mathematicians contributed to its solution. O. Bonnet himself showed in Bo] that besides the CMC surfaces there is a class of surfaces, depending on nitely many parameters, which allows non-trivial isometries preserving H. These results were developed further by L. Raay, who proved that the Bonnet surfaces are isothermic (i.e. allow conformal curvature line parametrization) and isometric to surfaces of revolution. J.N. Hazzidakis H] showed that the mean curvature function H satiies an ordinary diierential equation of the third order and was able to integrate it once. Graustein G] proved that all Bonnet surfaces are Weingarten surfaces, i.e. the 1 mean and the Gaussian curvature are related d H ^ d K = 0. He also found a convenient alternative description for the Bonnet surfaces. Namely, he showed that these surfaces can be characterized as isothermic surfaces, where the function 1=Q with Q = 1 4 < F xx ? F yy ; N > is harmonic, that means (@ xx + @ yy)1=Q = 0: In modern notations Q is the Hopf diierential, written in isothermic coordinates x, y. Later the problem was treated by E. Cartan in C], where the most detailed results concerning …
TL;DR: In this paper, the Gamma-limit of a rescaled version of an optimal material distribution problem for a cylindrical two-phase elastic mixture in a thin three-dimensional domain is explicitly computed.
Abstract: Abstract The Gamma-limit of a rescaled version of an optimal material distribution problem for a cylindrical two-phase elastic mixture in a thin three-dimensional domain is explicitly computed. Its limit is a two-dimensional optimal design problem on the cross-section of the thin domain; it involves optimal energy bounds on two-dimensional mixtures of a related two-phase bulk material. Thus, it is shown in essence that 3D-2D asymptotics and optimal design commute from a variational standpoint.