Showing papers in "Annals of Mathematics in 1990"

A proof of Langland’s conjecture on Plancherel measures; Complementary series of $p$-adic groups

Abstract: analysis of p-adic reductive groups. Our first result, Theorem 7.9, proves a conjecture of Langlands on normalization of intertwining operators by means of local Langlands root numbers and L-functions, at least when the group is quasi-split and the inducing representation is generic. Assuming two natural conjectures in harmonic analysis of p-adic groups, we also prove the validity of the conjecture in general (Theorem 9.5). As our second result we obtain all the complementary series and special representations of quasi-split p-adic groups coming from rank-one parabolic subgroups and generic supercuspidal represen

Interior $W^{2,p}$ estimates for solutions of the Monge-Ampère equation

Abstract: In this work we adapt the techniques developed in [C1] to prove interior estimates for solutions of perturbations of the Monge-Ampere equation

Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature

Abstract: We present a method for proving the existence of Kiihler-Einstein metrics of positive scalar curvature on certain compact complex manifolds, and use the method to produce a large class of examples of compact Kiihler-Einstein manifolds of positive scalar curvature. Suppose that M is a compact complex manifold of positive first Chern class. As is well-known, the existence of a Kiihler-Einstein metric on M is equivalent to the existence of a solution to a certain complex Monge-Ampere equation on M. To solve this complex MongeAmpere equation by the method of continuity, one needs only to establish the appropriate zeroth order a priori estimate. Suppose now that M does not admit a Kiihler-Einstein metric, so that the zeroth order a priori estimate fails to hold. From this lack of an estimate we extract various global algebro-geometric properties of M by introducing a coherent sheaf of ideals >J on M, called the multiplier ideal sheaf, which carefully measures the extent to which the estimate fails. The sheaf >Y is analogous to the "subelliptic multiplier ideal" sheaf that J. J. Kohn introduced over a decade ago to obtain sufficient conditions for subellipticity of the d-Neumann problem. Now >J is a global algebro-geometric object on M, and it so happens that >J satisfies a number of highly nontrivial global algebro-geometric conditions, including a cohomology vanishing theorem. In particular, the complex analytic subspace V c M cut out by >J is nonempty, connected, and has arithmetic genus zero. If V is zero-dimensional then it is a single reduced point, while if V is one-dimensional then its support is a tree of smooth rational curves. The logarithmic-geometric genus of M - V always vanishes. These considerations place nontrivial global algebro-geometric restric

The Iwasawa conjecture for totally real fields

Abstract: Let F be a totally real number field. Let p be a prime number and for any integer n let Fun denote the group of nth roots of unity. Let 41 be a p-adic valued Artin character for F and let F,, be the extension of F attached to 4, i.e., so that 4 is the character of a faithful representation of Gal(F,,/F). We will assume that F,, is also totally real. For a number field K let K., denote the cyclotomic Zp-extension of K. Following Greenberg we say that 4 is of type S if F., n Fc, = F and of type W if 4 is one-dimensional with F.,p c Fcc. Deligne and Ribet (in [DR], following Kubota and Leopoldt for the case F= Q) have proved the existence of a p-adic L-function associated to a one-dimensional Artin character 4 with F,, totally real. This function Lp(s, 4) is continuous for s e Zp{ 1}, and even at s = 1 if 4, is not trivial, and satisfies the following interpolation property:

A localization property of viscosity solutions to the Monge-Ampere equation and their strict convexity

Abstract: The purpose of this note is to show a localization property of convex viscosity solutions to the Monge-Ampere inequality 0 1−(2/n)) are strictly convex

Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature

Abstract: This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature problem, as well as to a number of models for phase transitions in two dimensions. We introduce a class of equations for which the initial value problem is solvable for initial data with p-integrable curvature, and we also give estimates for the rate at which the p-norms of the curvature must blow up, if the curve becomes singular in finite time. A detailed discussion of the way in which solutions can become singular and a method for "continuing the solution through a singularity" will be the subject of the second part.

Sandwiched singularities and desingularization of surfaces by normalized Nash transformations

Abstract: We prove that a succession of normalized Nash transformations resolves the singularities of a complex surface. We also give a classification of sandwiched surface singularities.

Green functions and character sheaves

Abstract: purely combinatorial way, using earlier ideas of P. Hall. In the general case, they were introduced [3] in terms of l-adic cohomology, as values of certain virtual representations of R'(O)(T is a maximal torus defined over Fq) at unipotent elements; the characters of RG( 0) at arbitrary elements were then expressible in a simple form in terms of Green functions of G and of smaller groups. One drawback of the definition of [3] was that it did not allow one to compute the Green functions except in some simple cases. Another definition of Green functions was proposed by Springer, first in terms of trigonometric sums on the Lie algebra and later in geometrical terms [17]. This definition was applicable only in large characteristic and in that case it was identified with the definition in [3] by Springer [17] and Kazhdan [10]. In [12], I observed that Green functions for GLn can be redefined in terms of intersection cohomology. This led me to a new definition of Green functions

Embedded minimal surfaces of finite topology

Abstract: In this paper we prove that any complete, embedded minimal surface M in R 3 with finite topology and compact boundary (possibly empty) is conformally a compact Riemann surface M with boundary punctured in a finite number of interior points and that M can be represented in terms of meromorphic data on its conformal completion M . In particular, we demonstrate that M is a minimal surface of finite type and describe how this property permits a classification of the asymptotic behavior ofM .

Symmetries of the CAR algebra

Abstract: A *-automorphism of period two of the UHF C*-algebra of type 2' is constructed for which the fixed-point algebra is not approximately finite dimensional. The construction also yields a diagonal maximal commutative C*-subalgebra whose spectrum is not totally disconnected. Variations and consequences of the construction are discussed.

Chow rings of moduli spaces of curves II: Some results on the Chow ring of $\overline{\mathscr{M}}_4$

Abstract: ring of the moduli space of stable curves of genus 4. These results are not complete. We find generators for the Chow ring of 4 and for the Chow groups in codimension 1 and 2 of -W4. For A2(G'4) we find fourteen generators. Using test surfaces we prove that the dimension of A 2(4/'4) is at least 13 and explicitly determine the single relation between the fourteen generators which still can exist. Finally, we have two proofs that this relation does indeed hold, so that the dimension of A2( 4/4) equals 13. This enables us to determine the Chow ring of ,4'4. Our original proof is based on a rather delicate argument; the second proof uses a result of Ran (see [R]) and is much simpler.

Invariant differential operators on hermitian symmetric spaces

Abstract: Our object of study is the ring of invariant differential operators on a hermitian symmetric space G/K of classical and noncompact type. Here, as usual, G is a connected noncompact semisimple Lie group with finite center and K is a maximal compact subgroup of G. It is well-known that the ring, denoted by ?(G/K), is isomorphic to a polynomial ring of 1 variables, 1 being the rank of G/K. This is true even in the nonhermitian case. If 1 = 1, it is generated by the Laplace-Beltrami operator z/ which is essentially self-adjoint; moreover -Iz2 is nonnegative. In the general case, an easily posable problem is to find an explicitly defined set of generators. We can go one step further by focusing our attention on the nonnegativity, which necessarily limits the range of eigenvalues under a suitable integrability condition on functions. It is thus natural to ask whether there exist some canonically defined operators which generate -9(G/K) and have the property of nonnegativity. The main purpose of the present paper is to give an affirmative answer to this question. In fact, our answer applies to a somewhat more general type of rings of differential operators which includes 9(G/K) as a special case. To be explicit, we take an irreducible representation p: K -* GL(V) with a complex vector space V of finite dimension, and consider the set C'(p) of all V-valued Cx functions f on G such that f(xk -1) = p(k)f(x) for every k e K. We then denote by ?(p) the ring of left-invariant difrrential operators on G which map C'(p) into itself. Now the complexification g of the Lie algebra of G has abelian subalgebras p+ and pwhich can be identified with the spaces of holomorphic and antiholomorphic tangent vectors on G/K at the origin. For any complex vector space W let Sr(W) denote the ring of all complex-valued homogeneous polynomial functions on W of degree r, and let S(W) = Er=0Sr(W). Through the adjoint representation of G on g, K acts naturally on Sr( ?+). It is a known fact, due to Hua and Schmid, that each irreducible constituent of this representation of K has multiplicity one. Now our principal

A restriction theorem for the Heisenberg group

Abstract: Since the discovery by E. M. Stein (see Fefferman [5]) that the Fourier transform of an LP-function on R' has a well-defined restriction to the unit sphere S`' which is square integrable on Sn-1, if p is close enough to 1, various new restriction theorems have been proved (see e.g. [10], [22], [27], [29]). The importance of such theorems in harmonic analysis (see e.g. [5], [6], [24], [28]) as well as in the theory of partial differential equations ([18], [19], [25]) has become evident. It has turned out [29] that the crucial step in proving the Stein-Tomas restriction theorem is the following: Let a denote the surface measure of S n 1, and let K denote the Fourier transform of a. Then the operator Af = f * K is bounded from LP to LP', if p is sufficiently close to 1. 9 can be interpreted in terms of the Laplacian A on Rn: Let -A = foX dEx be the spectral decomposition of A. Then, formally, 9 = const. f106(1 X) dEx, where 6 denotes Dirac's point measure. Therefore, it is natural to study analogues for 9, where A is replaced by a more general positive differential operator L. In the case of second order elliptic operators on compact manifolds, this has been done by Sogge [25].

Appendix to “The Iwasawa conjecture for totally real fields” by A. Wiles: Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces

Abstract: The arithmetic theory of toroidal compactification of the Hilbert-Blumenthal moduli spaces was described in Rapoport's thesis [Rap], while the arithmetic theory of minimal compactification has been lacking for a long time. After Faltings' work on the arithmetic compactification of Siegel moduli spaces [Fa], it was obvious to the experts that the same things could be done in the HilbertBlumenthal case in similar but simpler ways. (In fact, what emerged was a machinery to compactify moduli spaces "coming from abelian varieties".) It turns out that the arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces could have been done more than ten years ago. [Rap] contains everything that is needed, so in some sense I am writing a summary and an appendix to [Rap]. The purpose of this. appendix is to explain this remark, and show that this theory of minimal compactification can be useful in number theory. For instance, one can replace "for almost all primes" by "for all primes outside the level and the discriminant" in the main result of [Br-La]. Two more applications are given here, one about p-adic monodromy, the other about positivity of the Hodge line bundle (the sections of which are Hilbert modular forms). For a systematic exposition of the theory of semi-abelian degeneration and compactification, the reader is referred to the book [Fa-Ch], where the (more

On a conjecture of Brumer

Abstract: In a series of papers Iwasawa proposed and studied analogues for number fields of constructions due to Weil for curves over finite fields (cf. [11]). In particular he conjectured an analogue for Weil's theorem relating the zeta function of a non-singular curve over a finite field with the characteristic polynomial of its Frobenius automorphism. One strong piece of evidence for this came from the classical theorem of Stickelberger. However Iwasawa's original conjecture was proved for F= Q in [MW] without use of Stickelberger's theorem. For a general totally real number field Brumer proposed a generalization of Stickelberger's theorem. In [W2] we proved Iwasawa's conjecture for an arbitrary totally real field and here we use this to recover a version of Brumer's conjecture. We note however that Iwasawa's conjecture alone is not sufficient to prove our theorem owing to the phenomenon of trivial zeroes. The reason is that in Iwasawa theory one habitually interpolates the special values of Lfunctions which are imprimitive. The key part of this paper is the development of a technique, introduced in [W2], of adjoining an auxiliary variable to the p-adic b-functions in a way that permits the study of these primitive Lfunctions. I am grateful to the referee for correcting some obscure arguments. Let F be a totally real number field and let K be a CM field which is

Infinitesimal variation of Hodge structures and the weak global Torelli theorem for complete intersections

Abstract: In this paper, we will study the weak global Torelli theorem for complete intersections of hypersurfaces in a projective space. Let X be a non-singular projective variety of dimension r with an ample line bundle L over the complex number field C. Then the primitive part Hrm(X, Z) of the cohomology of X has a polarized Hodge structure. If the moduli space X of X exists, we can define the global period map Ho-* D/r from the moduli space X to the classifying space D/r of Hodge structures attaching the point p in X to the Hodge structure of the primitive part Hrim(Xp, Z) of the cohomology group of Xp(p E 4').

The differential zeta function for Axiom A attractors

Abstract: closed orbit of least period X( T). An important problem is to construct the meromorphic extensions for these zeta functions. A detailed knowledge of the domain of '(s) gives information on the long-term behaviour of the flow. For example, this is a standard approach to deriving asymptotic formulae for closed orbits; cf. [9]. For the case of geodesic flows on surfaces of constant negative curvature, the zeta function is known to have a meromorphic extension to the entire complex plane. Unfortunately, an obstruction to a general theory along these lines is the existence of examples of Axiom A flows for which D(s) does not have a meromorphic extension to C. In a recent article, William Parry proposed a modified definition of a zeta function for Axiom A attractors. He defined the differential zeta function by Ns(s) = H(I e-S(T)