Showing papers in "Inventiones Mathematicae in 1976"

The dirichlet problem for a complex Monge-Ampère equation

Abstract: where G' denotes the Jacobian determinant of G. Furthermore, if G4=(0 . . . . . 0), then (d d ~ log h G I) "= 0. Thus, for f = 0, (1) is a natural generalization of the Dirichlet problem for harmonic functions in the complex plane. Other extended Dirichlet problems were studied in connection with function theory in several variables by S. Bergman [2, 3] (on domains with distinguished boundary surfaces) and more generally by H. Bremermann [4]. In Section 8 it is shown that the solution of the problem discussed by Bremermann actually solves (1), in a generalized sense, with f = 0 . The problem (1) seems to be a reasonable candidate for a (nonlinear) potential theory associated with the theory of functions of several complex variables. The question of uniqueness for the problem (1) is related to the question of existence of "inner functions" on the domain O. If h is a bounded analytic function

B-Functions and Holonomic Systems. Rationality of Roots of B-Functions

Abstract: A b-function of an analytic function f(x) is, by definition, a gcnerator of the ideal formed by the polynomials b(s) satisfying P(s, x, Dx) f (x)\" + 1 = b(s) f(x) ~ for some differential operator P(s, x, Dx) which is a polynomial on s. Professor M.Sato introduced the notions of \"a-function\", \"b-function\" and \"'c-function\" for relative invariants on prehomogeneous vector spaces, when he studied the fourier transforms and ~-functions associated with them (see [10, 12]). He defined, in the same time, b-functions for arbitrary holomorphic functions and conjectured their existence and the rationality of their roots. Professor Bernstein introduced, independently of Prof. Sato, b-functions and proved any polynomial has a non zero b-function [1]. Professor Bj6rk extended this result to an arbitrary analytic functions by the same method [3]. The rationality of roots of b-functions is closely related to the quasi-uni-potency of local monodromy. In fact, Professor Malgrange showed that the eigenvalues of local monodromy are exp (2 nlfZ~a) for a root c~ of the b-function when f has an isolated singularity [9]. In this paper, the proof of the existence of b-functions and the rationality of their roots are given. The method employed here is to study the system of differential equations which satisfies f(x) ~. First, we will show that ~J~ is a subholonomic system and prove the existence of b-functions as its immediate consequence. Next, we study the rationality of roots of b-functions by using the desingularization theorem due to Hironaka. So, the main result of this paper is the following two theorems. Theorem, The characteristic variety oJ'~J ~ is equal to W s. Wf is, by dffinition, the closure of {(x, ~); ~ = s grad log f(x) for some sol2} in the cotangent vector bundle.

Zeta-functions for expanding maps and Anosov flows

Abstract: Given a real-analytic expanding endomorphism of a compact manifoldM, a meromorphic zeta function is defined on the complex-valued real-analytic functions onM. A zeta function for Anosov flows is shown to be meromorphic if the flow and its stable-unstable foliations are real-analytic.

Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus

Abstract: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 57 Chapter 1 : The Intersection Behaviour of the Curves T N . . . . . . 60 1.1. Special Points . . . . . . . . . . . . . . . . . . . . . 60 1.2. Modules in Imaginary Quadratic Fields . . . . . . . . . . 68 1.3. The Transversal Intersections of the Curves T N . . . . . . . 74 1.4. Contributions from the Cusps . . . . . . . . . . . . . . 78 1.5. Self-Intersections . . . . . . . . . . . . . . . . . . . . 82 Chapter 2: Modular Forms Whose Fourier Coefficients Involve Class Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.1. The Modular Form ~oo(z ) . . . . . . . . . . . . . . . . 88 2.2. The Eisenstein Series of Weight 3 . . . . . . . . . . . . . 91 2.3. A Theta-Series Attached to an Indefinite Quadratic Form . . 96 2.4. Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . I00 Chapter 3: Modular Forms with Intersection Numbers as Fourier Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 103 3.1. Modular Forms of Nebentypus and the Homology of the Hilbert Modular Surface . . . . . . . . . . . . . . . . . . . . t03 3.2. The Relationship to the Doi-Naganuma Mapping . . . . . 108 References . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Stratifications de Whitney et théorème de Bertini-Sard

Abstract: Apparatus for molding hollow bodies from plastic material by blow extrusion has a rotatable support with a plurality of successive, continguous molds arranged thereon, for rotation therewith. Each of the molds has a blowing device and an ejector associated therewith and arranged in the bottom portion of the mold preceeding the associated mold.

Completely Integrable Hamiltonian Systems Connected with Semisimple Lie Algebras

Abstract: In his recent paper [2] Moser considered such systems with potential V(q) = q- 2 (Calogero model [3]) and with V(q)=aZsinh-2aq (Sutherland model [4]). Using the Lax method [5] Moser has found in explicit form n independent integrals of motion. For the Calogero model he has also shown that these integrals are in involution (i.e. the Poisson brackets of any two integrals equal zero) and consequently according to Liouville theorem the system is completely integrable. The Moser's method was used to investigate analogous systems with potential V(q)=a z sinh -2 aq [6] and V(q)=a z go(aq) [7] (go(aq) is the Weierstrass func- tion). In paper [6] the corresponding quantum systems are also considered. In this paper we investigate the systems of the type (1.1) with potential U(q)=g 2 Z [V(qk--qt)+eV(qk+qt)] +g2 ~, V(qk)+g ~ ~ V(2qk) (1.3)

Root systems and elliptic curves

Abstract: This paper describes a new invariant theory for root systems. We briefly outline the main geometric applications; for precise results we refer the reader to the relevant theorems. Let R be a root system, R v its dual and QV the lattice generated by R v. If E is an elliptic curve over tr, then A : = QV | is an abelian variety on which the Weyl group W of R acts. It is easily shown that there is a W-invariant symmetric bilinear form I on QV such that 1 takes the value 2 on the roots of smallest length. It follows from the theorem of Appel l -Humbert that there exists a line bundle 5O over A of "Chern c lass"I such that the action of W on A lifts to an action of W on 5 ~ with the property that W leaves the fibre over 0cA fixed. Then 5 ~ 1 is ample and the algebra of W-invariant sections

The spectrum of Jacobi matrices

Abstract: Let L be a periodic symmetric tridiagonal matrix of size N; "periodic" means that L has one extra-entry in the upper right corner and by symmetry in the lower left one. Let b i be the diagonal and ai the subdiagonal entries. The present paper deals with the space ~ ' of such matrices with a given spectrum. On Jr' there is a natural class of commuting flows (isospectral deformations), which derive from Hamiltonian mechanics. When the given spectrum is non-degenerate, there are N 1 independent flows except for some degeneracies on some lower dimensional submanifolds. Each of these flows has in general N 1 integrals in involution, so that generically the solutions are quasi-periodic, their orbits are dense on a N 1 dimensional torus and there exists a canonical transformation to a set of action-angle variables. However there is much more involved, because these tori are algebraic surfaces and their periods can be expressed in terms of hyperelliptic functions; this is to say each such torus is a Jacobi variety. The transformation from the original variables (a~, bi) to a set of separation variables (/~i, vi) is of rational character. The "posi t ion" components p~ of these variables are provided by the spectrum of the matrix L, from which the first row and the first column has been removed. They define a local system of coordinates on the torus. Another system of coordinates t~ is provided by the group action of R N1 on the torus, such that the flows appear as linear motions on the torus. The Jacobi transformation maps the local system of coordinates (#1 . . . . . #N-l) into the global one (q . . . . . tN_l). The inverse map can be expressed in terms of the flows above and can be explicited in terms of quotients of theta functions invoking the theory of the Jacobi inversion problem. As a bonus, this yields explicit solutions to the differential equation defined by the isospectral flows, in terms of Abelian and theta functions. Finally, ~t' can be foliated by N-1-d imens iona l tori, each of which can be labelled by a modulus; this modulus is defined as the product of the non-diagonal

Spherical hypersurfaces in complex manifolds

Abstract: There has recently been a great deal of progress in the mapping or biholomorphic classication theory of domains with strongly pseudoconvex (s. ~. c.) boundaries in C"+l(n> 1). This is based on the combination of results of Chern-Moser [5] on the local differential geometry of real hypersurfaces in C" + 1, and C. Fefferman's marvelous extension theorem [6], which reduces the biholomorphic equivalence of s. ~k. c. domains to the CR equivalence of their boundaries. It is a simple consequence of this work that such domains have an infinite number of "moduli", i.e., there are smooth families of inequivalent such domains depending on arbitrarily many independent parameters (e.g., [3]). Given further restrictive assumptions on the local nature of the boundary, one expects to recover a finite-dimensional problem. In this paper, we make the most drastic assumption possible: we consider domains whose boundaries are everywhere locally CR equivalent to the unit sphere S 2n+l c[~n+l . Such hypersurfaces will be called spherical. The results are of two kinds. On the one hand, we classify the simply-connected spherical hypersurfaces M which are homogeneous under AutcR(M), the automorphism group of the CR structure on M. A consideration of compact space-forms associated to these homogeneous spaces shows the sphere and its quotients by roots of unity are the only compact, homogeneous spherical hypersurfaces. Together with a result of Webster, this completes the classification of compact homogeneous s.~O.c, hypersurfaces in [14]. On the other hand, it is not clear that any of the compact spherical hypersurfaces constructed in the course of the above bound domains in ~,+1, and we, at least, wondered whether S 2"+1 was the only example. In w we give a method for constructing large families of inequivalent such domains in ~"+ 1 Finally, we mention that some interesting global behavior of the chains of E. Cartan and Chern-Moser are observed in examples of w 6 and w 8. Rather more interesting and exotic behavior of chains has been found by Fefferman in [-7], in necessarily less symmetric boundaries than those considered here.