Showing papers in "Duke Mathematical Journal in 1980"

On the variety of special linear systems on a general algebraic curve

Abstract: 0. Introduction (a) Statement of the problem and of the main theorem; some references 233 (b) Corollaries of the main theorem 236 (c) Role of the Brill-Noether matrix 238 (d) Heuristic reasoning for the dimension count; the CastelnuovoSeveri-Kleiman conjecture 240 (e) Notations and terminology 242 1. Reduction of the dimension count to the conjecture (a) Heuristic discussion 244 (b) Geometry of Castelnuovo canonical curves 246 (c) Dimension count on Castelnuovo canonical curves 248 (d) Proof of the reduction theorem 249 2. Proof of the Castelnuovo-Severi-Kleiman conjecture (a) Heuristic discussion; the crucial examples 254 (b) Completion of the argument 259 3. Multiplicities of W, (a) Heuristic discussion 261 (b) Computation of an intersection number 263 (c) Solution to an enumerative problem on Castelnuovo canonical curves 266 (d) Determination of multiplicities for a general smooth curve 270

The mathematical theory of resonances whose widths are exponentially small

Abstract: It is shown how the rigorous justification of resonance widths in Paper I [5] can be simplified by exploiting Langer's trick of expanding the independent variable rather than the dependent variable [9].

Whittaker models for real groups

Abstract: Introduction. Whittaker functions were first introduced for the principal series representations of Chevalley groups by H. Jacquet [3]. Later, they were pursued by G. Schiffmann for algebraic groups of real rank one [12]. They played a very important role in the development of the Hecke theory for GLn through the work of H. Jacquet, R. P. Langlands, I. I. Piatetski-Shapiro, and J. A. Shalika [4, 5]. More precisely, they were the main tools for the definitions of local and global L-functions and e-factors. They also appeared quite useful in the development of the Hecke theory for other groups (cf. [10]), as well as in the definition of the local y-factors of certain functional equations [13, 14], particularly in their factorization. There seems to be other evidence of interest, especially in the work of W. Casselman, B. Kostant [7], and G. Zuckerman. The analytic behavior of these functions is much simpler when the ground field is non-archimedean; a good account of their analytic properties and some interesting formulas for certain class of such functions may be found in a recent paper of W. Casselman and J. A. Shalika [2]. But when the ground field is archimedean, these functions were believed to behave in a rather complicated manner. In fact, this has been one of the main obstacles in the development of the Hecke theory for number fields. To make a more precise statement of the problem, we let G be a split reductive algebraic group over R. We fix a maximal torus T of G and we let B be a fixed Borel subgroup of G containing T. We write B = M0AU, the Langlands decomposition of B with T=M0A, and fix a non-degenerate (unitary) character x of U (see section 1). Now, let IT be a continuous representation of G on a Frechet space V. Denote by ( oo > ^oo) e corresponding differentiable representation. Topologize V^ with the relative topology inherited from C°°(G, V). Let VK be the subspace of ^-finite vectors of F, where K is a fixed maximal compact subgroup of G with G = KB. We say that the representation (7r, V) is non-degenerate, if there exists a continuous linear functional A on V^, called a Whittaker functional, such that

Curvature characterization of hyperquadrics

Abstract: After proving the dimension two case jointly with Andreotti, Frankel [3] conjectured that a compact K/ihler manifold of positive sectional curvature is biholomorphic to the complex projective space. Mabuchi [8] verified the case of dimension three by using the result of Kobayashi-Ochiai [6]. Very recently by using the methods of algebraic geometry of positive characteristic Mori [10] proved that a compact Kihler manifold with ample tangent bundle must be biholomorphic to the complex projective space. By methods of Kihler geometry Siu-Yau [12] proved that a compact K/ihler manifold of positive holomorphic bisectional curvature must be biholomorphic to the complex projective space. Frankel’s conjecture is a special case of these more general results. It is reasonable to conjecture that there are similar curvature characterizations for other irreducible compact symmetric K/ihler manifolds. In this paper we obtain such a curvature characterization for the complex hyperquadric. Definition. Let M be a K/ihler manifold and P M. If the holomorphic bisectional curvature of M is nonnegative at P, then for a nonzero element (a) of the holomorphic tangent space T,M of M at P, the curvature null space at P in the direction of , denoted by N,(O, is defined as the set of all