Showing papers in "Mathematische Annalen in 1975"

On the cohomology groups of moduli spaces of vector bundles on curves

Abstract: A shaft seal of the type including an annular metal case and a flexible non-elastomeric sealing element such as polytetrafluoroethylene is made by molding a synthetic rubber filler ring between the metal case and the sealing element such that it bonds to both the metal case and to the sealing element, providing a faster and less expensive method of manufacturing the seal, while eliminating any leakage route between the metal case and the sealing element and also eliminating the I.D. to O.D. concentricity problem. In the embodiment where the sealing element is polytetrafluoroethylene, the filler ring chemically bonds to the metal case and mechanically bonds to the sealing element. Hydrodynamic pumping members can be molded onto the shaft engaging surface of the sealing element during the process of molding the synthetic rubber filler ring.

Sums involving the values at negative integers of L-functions of quadratic characters

Abstract: For each integer r > 1 we will define an arithmetic function H(r, N) which for r = 1 is the class number of (not necessarily primitive) quadratic forms of discriminant N and for r > 1 is essentially the value of ~K(r) where K = ~ ( ] / ~ ) . For r > 2 the function ~N>_0 H(r, N)e 2~i~ is a modular form of weight r + 1/2 on Fo(4 ). This implies numerous identities involving H(r, N). The analogous formulas for r = 1 (which do not follow from the methods of this paper) are classical "class number relations" of Kronecker, Hurwitz and others, as well as certain generalizations coming from the Selberg-Eichler trace formula and from recent work of Hirzebruch-Zagier. One of the tools used is of independent interest: given two modular f o r m s f a n d 9, there are certain bilinear expressions in the derivatives o f f and 9 which are again modular forms.

Stable principal bundles on a compact Riemann surface

Abstract: A radiator device adapted for attachment to a radiator pipe, comprising a mounting base wrapped about the radiator pipe and secured thereto by wire fastenings; a plurality of outwardly disposed radiating fins circularly positioned about said mounting base and extending outwardly therefrom. This radiating device is adapted to be installed upon any size flue pipe extending from a furnace to a chimney to transmit flue pipe heat to the surrounding atmosphere.

A Kronecker limit formula for real quadratic fields

Abstract: Then ((s, A) is (after analytic continuation) a meromorphic function of s with a simple pole at s = 1 as its only singularity. Moreover, the residue of ((s, A) at s = 1 is independent of the ideal class A chosen; this fact, discovered by Dirichlet (for the case of quadratic fields) is at the basis of the analytic determination of the class number of K. If we consider the Laurent expansion of ((s, A) at s = 1, however, say

The indecomposable representations of the dihedral 2-groups

Abstract: Let K be a field. We will give a complete list of the normal forms of pairs a, b of endomorphisms of a K-vector space such that a 2 b 2 = 0. Thus, we determine the modules over the ring R = K ( X , Y)/(X 2, y2) which are finite dimensional as K-vector spaces; here (X 2, y2) stands for the ideal generated by X 2 and y2 in the free associative algebra K (X, Y) in the variables X and Y. If G is the dihedral group of order 4q (where q is a power of 2) generated by the involutions 91 and 92, and if the characteristic of K is 2, then the group algebra K G is a factor ring of R, and the K G-modules KGM which have no non-zero projective submodule correspond to the K-vector spaces (take the underlying space of ~ M ) together with two endomorphisms a and b (namely multiplication by g ~ 1 and g 2 1 , respectively) such that, in addition to a Z b 2 -0 , also (ab) q = (ba) q = 0 is satisfied. We use the methods of Gelfand and Ponomarev developped in their joint paper on the representations of the Lorentz group, where they classify pairs of endomorphisms a, b such that ab = ba = O. The presentation given here follows closely the functorial interpretation of the Gelfand-Ponomarev result by Gabriel, which he exposed in a seminar at Bonn, and the author would like to thank him for many helpful conversations.

Poincaré polynomials of the variety of stable bundles

Abstract: §1. In this note, we indicate a few improvements to [3]. Let X be an irreducible, non-singular projective algebraic curve defined over a finite field Fq with q elements, of characteristic p. Let SL(n, d) be the coarse moduli scheme of isomorphism classes of stable vector bundles of rank n and determinant isomorphic to an lFq-rational line bundle L of degree d. [We will assume that (n, d) = 1.] By replacing lFq by a finite extension if necessary, we may assume that SL(n, d) is defined over IFq. As one might expect, it is then indeed true that the IFq-rational points of SL(n, d) are precisely the stable vector bundles on X defined over IFq (see [3]). By the Weil conjectures, it is easy to write down the Poincar6 polynomials of SL(n, d) once the number of its lFq-rational points is known. In order to compute the latter, one first notices that the fact that the Tamagawa number of SL(n) is 1 can be interpreted as follows.

A class of hypoelliptic pseudodifferential operators with double characteristics

Abstract: where II }l (~) is the usual Sobolev norm of derivatives of order < s in L 2 For operators of principal type one has fairly complete results concerning (11) so our main interest is to study double characteristics When Pm is non-negative and /~m-1 is purely imaginary at the characteristics, Radkevi~ [ 11] gave necessary and sufficient conditions for P to have the properties in question (For a proof see also Melin [13]) Starting from an example of Gru~in [5], Sj6strand [13] obtained such conditions when the characteristics form a symplectic manifold and P= vanishes precisely to the second order there A part of these results were also found independently by Boutet de Monvel and Trhves [2] A very interesting point here is that ~ _ t just has to avoid a discrete set of values, essentially the eigenvalues of a harmonic oscillator When the characteristics form a manifold which is either symplectic or involutive a construction of a parametrix in a class of pseudodifferential operators of type 1/2, 1/2 has been given by Boutet de Monvel [1], and Grigis [4] has extended the construction to characteristic manifolds where the symplectic form has constant rank, The case where the characteristic manifold is of codimension 2 is exceptional since the range of the Hessian of P,, at the characteristics may then be the whole complex plane We shall discuss this situation briefly in Section 6 just to show that it is not possible then to relax the hypotheses made by Sj/istrand [13, Theorem 12] that the manifold is symplectic and that the index of the Hessian is 2 In all the other results referred to above there is a proper closed convex cone (angle) FCtl; such that

Periodicity of branched cyclic covers

Abstract: Let K C S 2n+1 be a simple fibered knot, that is, an embedding of an (n -2 ) connected (2n-1)-manifold K in the (2n+ 1)-sphere whose complement fibers over S x with ( n 1)-connected fibers. (See 1.5 for the preCise definition.) In this paper we examine the periodicity, in k, of the smooth k-fold cyclic covers K k of S 2\"+ ~ branched along K. For K the trefoil knot in S 3, Fox [6, p. 192] discovered a homological periodicity of period 6, namely that H . ( K k + 6 ) ~ H . ( K k ) for all k>2. A second, apparently unrelated, example of such periodicity occurs in the links of Brieskorn singularities, where the link of the singularity Zo 3 + z 2 + . . . + z, z +z,+6t-~l for odd n_>_ 3 is diffeomorphic to a connected sum of ( 1) (\"+ ~)/2l copies of the Milnor sphere [1, p. 13]. We will show that these are special cases of the same phenomenon, namely a periodicity in the homology, homeomorphism, and diffeomorphism type of Kk when the manifold K is a rational homology sphere and the fibered knot K C S z\" + has periodic monodromy. In the first section we make precise the idea of a smooth branched cover. This has already been done in [5] for topological and simplicial branched covers; the analogous results in the smooth case are straightforward. Section2 proves the well-known result that the link of f ( zo , , z , ) k • . , Z n + X

Rationality of moduli spaces of stable bundles

Abstract: Let X be a complete non-singular algebraic curve of genus 9 > 2 over an algebraically closed field k, n a positive integer, d an integer and L a line bundle of degree d over X. It is well known that the isomorphism classes of stable bundles of rank n and determinant L over X form an irreducible non-singular quasiprojective variety S,,L(X) of dimension (n 2 1 ) (9-1) , which is projective if (n,d)= 1 (see [8, 10-13]). It is also easy to see that S,,L(X) is unirational. Our object in this paper is to prove

Extension theorems for reductive group actions on compact Kaehler manifolds

Abstract: It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent: