Showing papers in "Annals of Mathematics in 1978"

Instability in invariant theory

Abstract: Let V be a representation of a reductive group G. A fundamental theorem in geometric invariant theory states that there are enough polynomial functions on V, which are invariant under G, to distinguish disjoint closed G-invariant subsets of V. This result was given by D. Mumford in [11] for linearly reductive groups and conjectured by him to hold in general. Subsequently, it has been established for arbitrary reductive groups due to the efforts of C.S. Seshadri, M. Nagata and W. Haboush among others (see [15] and [12]). To use geometry to study invariants, for any closed G-invariant subset S of V, one may define a vector v in V to be S-unstable if the closure of the orbit G v meets S. In the classical case, S contains just the zero point O of V. The other fundamental theorem in geometric invariant theory states that v is S-unstable if and only if there is a one-parameter subgroup X of G such that v is S-unstable for the induced Gm-action on V. This result appears in Chapter 2 of Mumford in the classical case over an algebraically closed field. The above theorems, together with the numerical study of the action of the one-parameter subgroups of G, form the Hilbert-Mumford criterion for instability, which gives an effective means for finding all vectors v for which all invariants vanish (without actually finding any invariants!). In this paper, I will prove the second fundamental theorem for arbitrary S over a perfect ground field (Theorem 4-2). This solves a rationality question mentioned on page 64 of Mumford's book. For special ground fields, this has been done by D. Birkes in [1] for the real numbers and by M. Raghunathan in [13] for the algebraic number fields. In fact, M. Raghunathan uses Birkes' results and A. Borel's reduction theory to deduce his results. My proof of this theorem rests on the solution of another problem

Desingularization of two-dimensional schemes

Abstract: We present a new proof of the existence of a desingularization for any excellent surface (where "surface" means "two-dimensional reduced noetherian scheme"). The problem of resolution of singularities of surfaces has a long history (cf. the expository article [25]). Separate proofs of resolution for arbitrary excellent surfaces were announced by Abhyankar and Hironaka in 1967; to date (1977) full details have not yet been published (but cf. [2], [12], [13], [14] and [15]). Actually Hironaka's results on "embedded" resolution are stronger than what we shall prove, viz. the following theorem (which nevertheless suffices for many applications). Unless otherwise indicated, all rings in this paper will be commutative and noetherian, and all schemes will be noetherian and reduced. We say that a point z of a scheme Z is regular if the stalk 0,, of the structure sheaf at z is a regular local ring, and singular otherwise; Z is non-singular if all its points are regular.

Eigenfunctions of invariant differential operators on a symmetric space

Abstract: In his paper [14], S. Helgason conjectured that any joint-eigenfunction of all invariant differential operators on a symmetric space can be given by the Poisson integral. The purpose of this paper is to prove this conjecture (see corollary to the theorem in the Section 5). There have appeared several papers dealing with the conjecture in the case that the rank of the symmetric space is equal to one ([12], [13], [15], [17], [30], [34], [35], [36]). The proofs given in these papers follow an idea due to S. Helgason [14] and may be explained as follows. In the rank one case the algebra of all invariant differential operators is generated by the Laplace-Beltrami operator. First, one expands any eigenfunction of the laplacian into K-finite functions. Then these K-finite functions are also eigenfunctions of the laplacian and have boundary values in the natural way. The radial component of the laplacian gives rise to hypergeometric differential equations. Thanks to the classical results on hypergeometric functions, one can estimate the asymptotic behavior of the solution near the boundary. This enables us to prove that the sum of the boundary values of K-finite functions converges in the sense of analytic functionals. In the higher rank case, however, the radial components of invariant differential operators are not ordinary differential operators anymore, so that one is unable to apply the classical results. In the meanwhile, some of the present authors have recently generalized the notion of "regular singularity" for the ordinary differential equation to that for the system of partial differential equations. The essential point is that the system of invariant differential equations has regular singularity along the Martin boundary which assures the existence of the boundary values of a solution as "hyperfunctions." In the method mentioned above, one encounters the crucial difficulty in proving the exist-

On the Stickelberger ideal and the circular units of a cyclotomic field

Abstract: By a cyclotomic field, we shall mean a subfield of the complex numbers C generated over the rational numbers Q by a root of unity. Let k be an imaginary cyclotomic field. Let Cn = e2ri/" for any integer n > 1. There is then a unique integer m > 2, m t 2 mod 4, such that k Q(Qm); we call m the conductor of k. We consider in this paper two objects associated with k: the Stickelberger ideal S and the circular units C. Let G be the Galois group of k over Q, and let R = Z[G] be a group ring of G over the ordinary integers Z. S is then an ideal of R; C is a subgroup of the group of units Eof k. Let j denote the element of G induced by complex conjugation. If A is any G-module, we denote by A+ the set of elements a in A for which ja = a, and by Athe set of elements a in A for which ja= -a. Our main result will be a computation of the indices [R-: S-] and [E+: C+]. We now state our result precisely. Let h denote the class number of k, h+ the class number of k+ (the maximal totally real subfield of k), and put hh/h+. Then we prove the following.

The Kunze-Stein Phenomenon

Abstract: Si da una nuova dimostrazione di un, teorema di Kunze e Stein, che dice che, se 1≤p<2,L p(SL(2, R))*L 2(SL(2, R)) e contenuto inL 2(SL(2, R)). Questa nuova dimostrazione puo essere generalizzata per provare lo stesso teorema per ogni gruppo di Lie connesso, semisemplice, col centro finito.

Residues and zero-cycles on algebraic varieties

Abstract: I. Residue theorem and interpretations . 466 a) Local properties of residues 466 b) The residue theorem and a converse 467 c) Cayley-Bacharach property and multisecant varieties . .. 470 d) Points and line bundles on curves . 471 e) Points and rank-two bundles on surfaces . 474 II. Residues and the osculating sequence . 476 a) The osculating sequence 476 b) The fundamental relation 478 c) The fundamental bound for complete intersections ...... 480 d) The osculating sequence for curves 484 e) The osculating sequence for surfaces 486 III. Inverting the residue theorem 490 a) Complete intersections on surfaces 490 b) Structure of extremal varieties, i) 494 c) Structure of extremal varieties, ii) 496 Appendix: Some observations and open problems ... 502

The Distance in BMO to L

Abstract: (9dx T I -III XI~ is the mean of cp over I. Every bounded (measurable) function is in BMO and IIpII* 0 and x0 =x(s) such that (1.2) sup, 1 | {x G I: I 9(x) -' l > A} I < e-/'

The Structure of Countable Boolean Algebras

Abstract: A structure theory for denumerable Boolean algebras is presented. The isomorphism types of countable Boolean algebras are completely characterized in terms of certain naturally defined invariants. As an application, we show that any countable commutative semigroup can be embedded into the semigroup of the isomorphism types of all countable Boolean algebras under direct product. This solves in particular the cube problem of Tarski.

Codimension one foliations of closed manifolds

Abstract: The global solutions (integral manifolds) of a completely integrable Pfaffian system are in general very intricately interwoven. Indeed, the appropriate language with which to describe a given foliation (i.e., the family of solutions of such a system) has not been fully developed, since the range of possibilites is not entirely known. This paper provides part of the information needed for a global classification, up to a diffeomorphism preserving the foliation, in the case where the n-manifold M on which the system (equation) is defined is compact without boundary and the dimension of the integral manifolds (leaves) is (n - 1). Euler begins the third volume of his Institutionum Calculi Integralis [3] with a treatment of global singular codimension one foliations on R3. His starting point is that when certain one-forms =Pdx + Qdy + Rdz are multiplied by functions M they become exact differentials d V, and their "complete integrals" are obtained by setting V equal to constants. He first observes that Mdcv - (v A dM 0, and then eliminates M and its derivatives by computing GO A do 0. This latter condition, written explicitly as

Closed geodesics and the y-invariant

Abstract: In [3], Atiyah, Patodi and Singer introduced an invariant of a Riemannian manifold of dimension 4n 1. This invariant, which they called the p-invariant, is determined by the spectrum of a certain self-adjoint square root of the Laplacian on differential forms. It is non-local; that is, it is not obtained by integrating a universal polynomial in curvature over the manifold. Thus, unlike earlier invariants determined by the spectrum such as the Euler characteristic or the signature, it cannot be computed from the asymptotic expansion of Trace-t, the trace of the heat operator, as t goes to zero. However, recently, Colin de Verdiere [26], Chazarain [7] and Duistermaat-Guillemin [9] discovered a connection between the spectrum of the Laplacian and non-local information about a Riemannian manifold by studying the distribution trace of the fundamental solution of the wave equation. For generic manifolds the singularities of this distribution on the real line are at the set of lengths of closed geodesics and there is an asymptotic expansion at each singularity with coefficients giving information about the closed geodesic. It is an important question to decide if the 72-invariant can be determined from these data. The following formula shows this is indeed the case for manifolds of constant negative curvature and suggests a general formula. Let M be a compact oriented 4n 1 dimensional Riemannian manifold of constant negative curvature. Let $I be the set of primitive closed geodesics on M. Then each y C 9I determines the holonomy element R(7) e SO(4n 2) by parallel translation around y, the (linearized) Poincare map P(-i) C Sp (8n 4, R) and the length L('Y) of J. We stop to give a definition of P(7). Let q', denote the geodesic flow on S(M), the unit tangent bundle of M. Then a closed geodesic of length L corresponds to a fixed-point of pL. Then dpL maps the tangent space of that fixed-point to itself and preserves the geodesic flow direction and hence induces a transformation P(I) normal

On vector fields as generators of flows: A counterexample to Nelson's conjecture

Abstract: A construction is given for a collection of examples of divergenceless vector fields with integral curves along which, in a finite time, fibers are contracted to points. The examples, which include a uniformly bounded vector field, are used to prove that, contrary to a conjecture due to E. Nelson, no LP condition on the vector fields is sufficient, in Rd, d 2 3, for the existence or uniqueness of a generated measure preserving point flow.

The arithmetic of automorphic forms with respect to a unitary group

Abstract: utilized in various arithmetical problems as well as in the study of the analytic properties of the form itself. The same can be said also for the Hilbert and Siegel modular forms. One expresses a given modular form F as a function of complex variables u,, * * *, u. with an expansion (0.1) F(u,, *. Us) = Ex c exp(2wi. * 1U'), where the coefficients c. are complex numbers and x runs over a lattice. Especially important are those F for which all c, are algebraic, or more restrictedly, cyclotomic. They form a distinguishable class which is stable under the transformation by the elements of the algebraic group in question. The algebraicity of c., is also indispensable if the value of a modular function at a special point is the problem, as in the theory of complex multiplication. In general, an expansion of type (0.1) exists for an automorphic form if it is defined on a tube domain and the group contains sufficiently many translations. There are, however, cases in which no such expansion is available. A typical example is provided by the symmetric domain

Uniform approximation by functions analytic on a Riemann surface

Abstract: The concern of this paper is the following approximation problem: if E is a closed subset of an open Riemann surface M, (under what conditions) can every function which is continuous on E and analytic on the interior of E be approximated uniformly on E by functions which are analytic on all of M? It will be necessary to define a few terms before stating the most general result, but the following special case includes many situations of interest.

Independent variation of secondary classes

Abstract: teristic classes of a certain class of foliations. The examples we compute show that many of these secondary classes vary linearly independently. This generalizes a result due to Thurston [T] on the variation of one of these classes, the Godbillon-Vey class. It is also a partial analogue of the results of Bott [B2] and Lazarov-Pasternack [LP] on the independent variation of the secondary classes for holomorphic and Riemannian foliations respectively. We are also able to compute the Simons' characters of these foliations and we show that many of them vary linearly independently. The method we use is a generalization of the theory of residues of singular foliations due to Baum-Bott [BB]. We work with the natural foliation z on a flat vector bundle and a vector field X tangent to the fiber of the bundle which preserves the foliation. We assume the vector field has an isolated singularity along the zero section. This situation determines certain cohomology classes on the zero section, the residues of z- and X. We relate these residues to the secondary characteristic classes and Simons' characters of the foliation ' spanned by z and X off the zero section. Using this technique we compute examples which show that the residues of Z and X and thus the characteristic classes of Z vary linearly independently. In Section 2 we record some facts we will use later. In Section 3 we