Showing papers in "Mathematische Annalen in 1993"

Universal enveloping algebras of Leibniz algebras and (co)homology

Abstract: The homology of Lie algebras is closely related to the cyclic homology of associative algebras [LQ]. In [L] the first author constructed a \"noncommutative\" analog of Lie algebra homology which is, similarly, related to Hochschild homology [C, L]. For a Lie algebra g this new theory is the homology of the complex C,(g) ... ~ ~| g|-+ ... ~1 ~ k, whose boundary map d is given by the formula d(gl|174 = ~ (-1)J(gl@'\"|174174174 \" l

New bounds in some transference theorems in the geometry of numbers

Abstract: The aim of this paper is to give new bounds in certain inequalities concerning mutually reciprocal lattices in R". To formulate the problem, we have to introduce some notation and terminology. We shall treat IR" as an n-dimensional euclidean space with the norm II II and metric d. The inner product of vectors u, v will be denoted by uv; we shall usually write u 2 instead of uu. The closed and open unit balls in F," will be denoted by B, and B'., respectively. If A c IR ", then span A and A z denote the linear subspace spanned over A and the orthogonal complement of A in P~". A lattice in IR" is an additive subgroup of IR" generated by n linearly independent vectors. The family of all lattices in IR" will be denoted by A.. Given a lattice L e A., we define the polar (dual, reciprocal) lattice L* in the usual way: L* = {u ~ ~ " : u v ~ Z for each v ~ L } .

Homotopy Types of Subspace Arrangements via Diagrams of Spaces.

Abstract: We prove combinatorial formulas for the homotopy type of the union of the subspaces in an (affine, compactified affine, spherical or projective) subspace arrangement. From these formulas we derive results of Goresky & MacPherson on the homology of the arrangement and the cohomology of its complement. The union of an arrangement can be interpreted as the direct limit of a diagram of spaces over the intersection poset. A closely related space is obtained by taking the homotopy direct limit of this diagram. Our method consists in constructing a combinatorial model diagram over the same poset, whose homotopy limit can be compared to the original one by usual homotopy comparison results for diagrams of spaces.

On b-function, spectrum and rational singularity

Abstract: by duality [18], because R~Tr,CCy, = 0 for i > 0 by [6, 31] (this follows also from [13, 21, 23]) where rc is assumed projective. Here COy denotes the dualizing sheaf (i.e., the dualizing complex [18] shifted by the dimension to the right). The trace morphism (0.2) is injective, and its image is independent of the choice of resolution, because (0.2) is an isomorphism if Y is smooth. We will denote by CSy the image of (0.2). See (2.4.8) below. Now assume Y is a reduced divisor D on a complex manifold X of dimension n. Let f be a reduced defining equation of D on a neighborhood of a: C D. The b-function (i.e., Bernstein polynomial) of f is a monic polynomial bf(s) in s with rational coefficients, and is a generator of the ideal whose element b(s) satisfies the relation

On the signature of four-manifolds with universal covering spin

Abstract: In this note we study closed oriented 4-manifolds whose universal covering is spin and ask whether there are restrictions on the divisibility of the signature. Since any natural number appears as the signature of a connected sum of r 2,s, without the assumption on the universal covering there cannot exist any restrictions. Certainly, the most famous such restriction was proved by Rohlin in [10], where he showed that the signature a of a smooth 4-dimensional spin manifold is divisible by 16 (compare part (2) of our Main Theorem for a new proof). The Kummer surface K shows that this is the best possible general result. Dividing by a certain free holomorphic involution on K, one obtains the Enriques surface (compare [1]) which by construction has signature 8 and fundamental group 7//2. Furthermore, Hitchin showed in [5] that there exists an antiholomorphic free involution on the Enriques surface. We will refer to the quotient as the Hitchin manifold which then has signature 4 and fundamental group 7//2 x 7//2. Rohlin's theorem admits a nice generalization to nonspin 4-manifolds, compare [4, Theorem 6.3]:

Random polytopes in a convex polytope, independence of shape, and concentration of vertices

Abstract: Write ~.-"d for the set of all convex bodies (convex compact sets with nonempty interior) in ~d. Define o@g~l d as the set of those K E 5 b "~d with vol K = 1. Fix K E .~g-i d and choose points X l , . . . , x~ E K randomly, independently, and according to the uniform distribution on K. Then K,~ = c o n v ( x l , . . . , xn} is a random polytope in K . Write E(K, n) for the expectation of the random variable v o l ( K \ K n ) . E(K, n) shows how well K,~ approximates K in volume on the average. Groemer [Grl] proved that, among all convex bodies K E o@g~l d, the ellipsoids are approximated worst, i.e.

Über Bewertungen, welche unter einer reduktiven Gruppe invariant sind

Abstract: LetG be a reductive group defined over an algebraically closed fieldk and letX be aG-variety. In this paper we studyG-invariant valuationsv of the fieldK of rational functions onX. These objects are fundamental for the theory of equivariant completions ofX. LetB be a Borel subgroup andU the unipotent radical ofB. It is proved thatv is uniquely determined by its restriction toKU. Then we study the set of invariant valuations having some fixed restrictionv0, toKB. Ifv0 is geometric (i.e., induced by a prime divisor) then this set is a polyhedron in some vector space. In characteristic zero we prove that this polyhedron is a simplicial cone and in fact the fundamental domain of finite reflection groupWX. Thus, the classification of invariant valuations is almost reduced to the classification of valuations ofKB.

On the topology of double coset manifolds

Abstract: We consider a compact Lie group G and two closed subgroups H and K of G. The abstract product K x H operates on G by g. (k, h) = k lgh. If this operation happens to be free, the quotient K \ G / H is a compact manifold, which we call a double coset manifold. These manifolds have attracted the attention of several differential geometers: Gromol] and Meyer [GrM] for instance described an exotic 7-sphere as a double coset manifold. Therefore the class of double coset manifolds is strictly larger than that of homogeneous spaces since Borel observed long ago that a homogeneous space which is homeomorphic to a sphere is actually diffeomorphic to it. A serious study of double coset manifolds was made by Eschenburg [E1-E4]. He showed that some of them admit Riemannian metrics with strictly positive sectional curvature; he obtained a classification of certain types of double coset manifolds and computed the cohomology of some of them. The objective of the present paper is to show that in many respects the topology of double coset manifolds is as easy to handle as that of homogeneous spaces. To compute the cohomology of G/H, the best way is to look at the fibration

Some uniqueness and nonexistence theorems for embedded minimal surfaces

Abstract: Thanks to the works of Callahan, Costa, Hoffman, Karcher, Meeks, Rosenberg, Wei, etc.—see for example [1],[2],[3],[5],[8],[11],[12],[16],[20]—, we dispose now of a large number of properly embedded minimal surfaces in the euclidean space IR other than the classical examples. All those surfaces can be viewed as minimal surfaces with finite total curvature properly embedded in complete flat three manifolds. The most basic invariants associated to a surface of this type are its topology and its periodicity. It is an interesting problem to decide if the simplest examples—like the catenoid, the helicoid, the Scherk’s surfaces or the Riemann example—can be characterized in terms of some of these invariants. In the non-periodic case, there are two important uniqueness theorems in this direction: the first one, obtained by Schoen [17], characterizes the catenoid as the only complete minimal surface embedded

On zeroes of the twisted tensor L-function

Abstract: A. Notations, results, remarks. Let E=F be a separable quadratic extension of global elds, A = A F and A E the associated rings of adeles, and A ; A E their multiplicative groups of ideles. Signify by G the group scheme GL(n) over F , and put G = G(F ), G = G(E), G = G(A ), G 0 = G(A E ), and Z(' F ), Z (' E );Z(' A ), Z(' A );Z(' A E ) for their centers. Fix a unitary character " of Z=Z , and denote by a cuspidal G 0 -module whose central character is ". Such a is called distinguished if there is a form in such that R

Unramified cohomology and rationality problems

Abstract: — The aim of this paper is to construct unirational function fields K over an algebraically closed field of characteristic 0 such that the unramified cohomology group H i nr(K,μ ⊗i p ) is not trivial for i = 2, 3 or 4 and p a prime number. This implies that the field K is not stably rational. For this purpose, we give a sufficient condition for an element to be unramified in H i(K,μ⊗i p ). This condition relies on computations in the exterior algebra of a vector space of finite dimension over the finite field Fp. Among the first examples of smooth projective varieties X over C which are unirational but not rational was the example constructed by Artin and Mumford using the torsion part of H3(X,Z). When X is unirational, this group may also be described as the unramified Brauer group of the function field of X . From this point of view, Saltman [Sa] and Bogomolov [Bo] gave examples related to Noether’s problem. Colliot-Thélène and Ojanguren [CTO] were the first to use the unramified cohomology groups in degree 3 to prove the non-rationality of a unirational field. The plan of this paper is the following: first we recall some basic facts about unramified cohomology. In the second section, we state the main result, Theorem 2, which enables one to characterize unramified elements by calculations in the exterior algebra. This generalizes some of the methods used in [Sa] and [Bo]. In the next section, we prove Theorem 2. In this proof, we show how one can lift the residue map in the exterior algebra of a subgroup of H(K,μp) of finite dimension. The fourth section applies the main result to the construction of several unirational non-rational fields. In this part, to prove the non-triviality of elements in H3 nr(K,μ ⊗3 p ), we use a recent result by Suslin [Su] and to have a ∗Math. Ann 296 (1993), 247–268