Showing papers in "Annals of Mathematics in 1997"

Quotients by groupoids

Abstract: We show that if a flat group scheme acts properly, with finite stabilizers, on an algebraic space, then a quotient exists as a separated algebraic space. More generally we show any flat groupid for which the family of stabilizers is finite has a uniform geometric, uniform categorical quotient in the category of algebraic spaces. Our argument is elementary and essentially self contained.

Heisenberg algebra and Hilbert schemes of points on projective surfaces

Abstract: The purpose of this paper is to throw a bridge between two seemingly unrelated subjects. One is the Hilbert scheme of points on projective surfaces, which has been intensively studied by various people (see e.g. [I], [Br], [ES], [G61], [G62]). The other is the infinite dimensional Heisenberg algebra which is closely related to affine Lie algebras (see e.g. [K]). We shall construct a representation of the Heisenberg algebra on the homology group of the Hilbert scheme. In other words, the homology group will become a Fock space. The basic idea is to introduce certain "correspondences" in the product of the Hilbert scheme. Then they define operators on the homology group by a well-known procedure. They give generators of the Heisenberg algebra, and the only thing we must check is that they satisfy the defining relation. Here we remark that the components of the Hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components. Our correspondences live in the product of the different components. Thus it is quite essential to study all components together. Our construction has the same spirit as the author's construction [Nal], [Na4] of representations of affine Lie algebras on homology groups of moduli spaces of "instantons"1 on ALE spaces which are minimal resolutions of simple singularities. Certain correspondences, called Hecke correspondences, were used to define operators. These twist instantons along curves (irreducible components of the exceptional set), while ours twist ideals around points. In fact, the Hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles, or more precisely torsion free sheaves. Our construction should be considered as a first step to extend [Nal], [Na4] to general 4-manifolds. The same program was also proposed by Ginzburg, Kapranov, and Vasserot [GKV].

Ricci curvature and volume convergence

Abstract: The purpose of this paper is to give a new (integral) estimate of distances and angles on manifolds with a given lower Ricci curvature bound. This will provide us with an integral version of the Toponogov comparison triangle theorem for Ricci curvature and "almost extreme triangles" (see the earlier works [Cl] and [C2] for an analog of this when the manifold has positive Ricci curvature). Using this, we prove the Anderson-Cheeger conjecture saying that the volume is a continuous function on the space of all closed n-manifolds with Ricci curvature greater or equal to -(n - 1) equipped with the GromovHausdorff metric. We also prove Gromov's conjecture (for n 57 3) saying that an almost nonnegatively Ricci curved n-manifold with first Betti number equal to n is a torus. Further, we prove a conjecture of Anderson-Cheeger saying that an open n-manifold with nonnegative Ricci curvature whose tangent cone at infinity is in is itself in. Finally we prove a conjecture of Fukaya-Yamaguchi. We will now describe these results in more detail. Let dGH denote the Gromov-Hausdorff distance [GLP]. First we have the following result which was conjectured by Anderson-Cheeger.

Connecting invariant manifolds and the solution of the $C^1$ stability and $\Omega$-stability conjectures for flows

Abstract: There is a gap in the proof of Lemma VII.4 in [Ann. of Math. (2) 145 (1997), 81--137]. We present an alternative proof of Theorem B (C^1 Omega-stable vector fields satisfy Axiom A). The novel and essential part in the proof of the stability and Omega-stability conjectures for flows is the connecting lemma introduced previously. A mistake in the proof of the last conjecture was pointed out to me by Toyoshiba, who later also provided an independent proof of it, again based on the connecting lemma and previous arguments by Ma\\~n\\'e and Palis.

On random matrices from the compact classical groups

Abstract: If M is a matrix taken randomly with respect to normalized Haar measure on U(n), O(n) or Sp(n), then the real and imaginary parts of the random variables Tr(Mk), k > 1, converge to independent normal random variables with mean zero and variance k/2, as the size n of the matrix goes to infinity. For the unitary group this is a direct consequence of the strong Szeg6 theorem for Toeplitz determinants. We will prove a conjecture of Diaconis

Scattering asymptotics for Riemann surfaces

Abstract: In this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends. We also give Weyl asymptotics for the relative scattering phase of such a surface. The proofs are based on trace formulae analogous to those of the Euclidean odd-dimensional scattering. The main technical ingredient is a new proof of the Poisson formula (Theorem 5.7) which is applicable in the Euclidean case as well. Our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries. The previous general lower bounds or asymptotics were either nonoptimal ([25], [58], [9]), one-dimensional or radial ([65], [67] and [54], [41]1) or they required some degeneracy of the

Cyclic Splittings of Finitely Presented Groups and the Canonical JSJ-Decomposition

Abstract: The classification of stable actions of finitely presented groups on ℝ-trees has found a number of applications Perhaps one of the most striking of these applications is the theory of canonical JSJ-decompositions of hyperbolic groups developed by Sela in his seminal paper [Se]

A priori estimates for prescribing scalar curvature equations

Abstract: We obtain a priori estimates for solutions to the prescribing scalar curvature equation (1) R(x)n-2 on Sn for n > 3. There have been a series of results in this respect. To obtain a priori estimates people required that the function R(x) be positive and bounded away from 0. This technical assumption has been used by many authors for quite a few years. It is due to the fact that the standard blowing-up analysis fails near R(x) = 0. The main objective of this paper is to remove this well-known assumption. Using the method of moving planes, we are able to control the growth of the solutions in the region where R is negative and in the region where R is small, and thus obtain a priori estimates on the solutions of (1) for a general function R which is allowed to change signs.

Ergodic theory on moduli spaces

Abstract: Let M be a compact surface with x(M) < 0 and let G be a compact Lie group whose Levi factor is a product of groups locally isomorphic to SU(2) (for example SU(2) itself). Then the mapping class group rM of M acts on the moduli space X(M) of flat G-bundles over M (possibly twisted by a fixed central element of G). When M is closed, then FM preserves a symplectic structure on X(M) which has finite total volume on M. More generally, the subspace of X(M) corresponding to flat bundles with fixed behavior over AM carries a rM-invariant symplectic structure. The main result is that rM acts ergodically on X(M) with respect to the measure induced by the symplectic structure.

Yang's system of particles and Hecke algebras

Abstract: The graded Hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions. This operator algebra arises from the study of the root system analogue of Yang's system of n particles on the real line with delta function potential. It turns out that the spectral problem for this generalization of Yang's system is related to the problem of finding the spherical tempered representations of the graded Hecke algebra. This observation turns out to be very useful for both these problems. Application of our technique to affine Hecke algebras yields a simple formula for the formal degree of the generic Iwahori spherical discrete series representations.

Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus

Abstract: Let f be a local homeomorphism of the plane with a fixed point z which is a locally maximal invariant set and which is neither a sink nor a source. We prove that there are two integers q > 1 and r > 1 such that the sequence i(fk, Z) of the indices at z of the iterates of f satisfy i(fk, z) = 1 - rq if k is a multiple of q and i(fk, z) = 1 otherwise. As a corollary we deduce that there is no minimal homeomorphism on the infinite annulus or more generally on the two-dimensional sphere minus a finite set of points. We also construct for a local homeomorphism f as above a topological invariant which is a cyclically ordered set with an automorphism on it; this allows us in particular to define a rotation number for f (rational of denominator q).

Quotient spaces modulo algebraic groups

Abstract: The paper proves that if a reductive group scheme acts properly on a scheme then the geometric quotient exists as an algebraic space. As a consequence we obtain the existence of the moduli spcace of canonically polarized varieties over Spec Z.

The Ehrhart polynomial of a lattice polytope

Abstract: The problem of counting the number of lattice points inside a convex lattice polytope in R' (a polytope whose vertices have integer coordinates) has been studied from a variety of perspectives. Here we use the Poisson summation formula and related techniques in Fourier analysis to obtain a formula for the number of lattice points inside all integral dilates of a lattice polytope. We show that an associated generating function has an explicit representation in terms of sums of cotangents, which are higher-dimensional generalizations of Dedekind sums. Let En denote the n-dimensional integer lattice in R', and let P be an n-dimensional lattice polytope in R', which is a compact simplicial complex of pure dimension n whose vertices lie on the lattice. Consider the function of an integer-valued variable t that describes the number of lattice points that lie inside the dilated polytope tP:

Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds

Abstract: The central result of this paper is a generalization of the Ehresmann fibration theorem to the infinite-dimensional and/or non-proper setting. With this aim, we introduce the concepts of strong submersion and of mapping with uniformly split kernels. First applications include a global Implicit Function Theorem and a necessary and sufficient version of Hadamard's global invertibility criterion in the setting of Finsler manifolds. Another major application makes use of the idea of asymptotic critical value (not critical point), which helps formulate various generalizations of the Palais-Smale condition for morphisms of Finsler manifolds, not merely functionals. We obtain a critical point theory for morphisms of Finsler manifolds extending many results known only for functionals, notably Ekeland's "variational" principle. These results are used to discuss a new approach to Lagrange multiplier and nonlinear eigenvalue problems, and to develop an intrinsic critical point theory for complex-analytic functionals. The latter reveals an intimate connection between conditions of Palais-Smale type and the structure of polynomial automorphisms (Jacobian Conjecture).

The most continuous part of the plancherel decomposition for a reductive symmetric space

Abstract: We prove the Plancherel formula for spherical Schwartz functions on a reductive symmetric space Our starting point is an inversion formula for spherical smooth compactly supported functions The latter formula was earlier obtained from the most continuous part of the Plancherel formula by means of a residue calculus In the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized Eisenstein integrals and of the Maass Selberg relations satis ed by the associated C functions

Localization and completion theorems for mu-module spectra

Abstract: Let G be a finite extension of a torus. Working with highly structured ring and module spectra, let M be any module over MU ; examples include all of the standard homotopical MU -modules, such as the Brown-Peterson and Morava K-theory spectra. We shall prove localization and completion theorems for the computation of M∗(BG) and M∗(BG). The G-spectrum MUG that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum SG, and there is an MUG-module MG whose underlying MU -module is M . This allows the use of topological analogues of constructions in commutative algebra. The computation ofM∗(BG) andM∗(BG) is expressed in terms of spectral sequences whose respective E2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring MU ∗ and its module M ∗ . The central feature of the proof is a new norm map in equivariant stable homotopy theory, the construction of which involves the new concept of a global I∗-functor with smash product.

A rigid analytic Gross-Zagier formula and arithmetic applications

Abstract: Let f be a newform of weight 2 and squarefree level N. Its Fourier coefficients generate a ring Of whose fraction field Kf has finite degree over Q. Fix an imaginary quadratic field K of discriminant prime to N, corresponding to a Dirichlet character E. The L-series L(f /K, s) = L(f, s)L(f 0 E, s) of f over K has an analytic continuation to the whole complex plane and a functional equation relating L(f/K, s) to L(f/K, 2 s). Assume that the sign of this functional equation is 1, so that L(f/K, s) vanishes to even order at s = 1. This is equivalent to saying that the number of prime factors of N which are inert in K is odd. Fix any such prime, say p. The field K determines a factorization N = N+Nof N by taking N+, resp. Nto be the product of all the prime factors of N which are split, resp.

Quantum limits on flat tori

Abstract: We classify all weak * limits of squares of normalized eigenfunctions of the Laplacian on two-dimensional flat tori (called quantum limits). We also obtain several results about such limits in dimensions three and higher. Many of the results are a consequence of a geometric lemma which describes a property of simplices of codimension one in R^n whose vertices are lattice points on spheres. The lemma follows from the finiteness of the number of solutions of a system of two Pell equations. A consequence of the lemma is a generalization of the result of B. Connes. We also indicate a proof (communicated to us by J. Bourgain) of the absolute continuity of the quantum limits on a flat torus in any dimension. After generalizing a two-dimensional result of Zygmund to three dimensions, we discuss various possible generalizations of that result to higher dimensions and the relation to L^p norms of densities of quantum limits and their Fourier series.

Infinitely many periodic attractors for holomorphic maps of $2$ variables

Abstract: An important development in the study of discrete dynamical systems was Newhouse's use of persistent homoclinic tangencies to show that a large set of C2 diffeomorphisms of compact surfaces have infinitely many coexisting periodic attractors, or sinks [6], where "large" refers to a residual subset of an open set of diffeomorphisms. In the present paper, we obtain this result for various spaces of holomorphic maps of two variables. Newhouse later extended his result to show that such residual sets exist near any surface diffeomorphism which has a homoclinic tangency [7]. More recently, Palis and Viana extended this latter result to higher dimensions when the stable manifold has codimension one [10], and Romero obtained an analogous result for higher codimension stable manifolds using saddles in place of sinks [12]. In each case, however, the construction reduces to the study of intersecting Cantor sets in the line: under an appropriate projection, the stable and unstable manifolds of a basic set are mapped to Cantor sets in the line, and a tangency between these manifolds corresponds to a point of intersection of the Cantor sets. The generic unfolding of tangencies then gives rise to periodic attractors or saddles. This reduction to linear Cantor sets depends heavily on the fact that there is only one expanding eigenvalue. Even Romero's result for higher codimension stable manifolds involves a reduction to this case. In the holomorphic setting, eigenvalues come in conjugate pairs (from a real point of view), so this reduction is not possible. Instead, stable and unstable manifolds are Riemann surfaces, and after extending the stable and unstable manifolds of a basic set to foliations, these foliations will be tangent in a (real) 2-dimensional disk, and the stable and unstable manifolds correspond to Cantor sets in this disk. Hence, persistent tangencies between basic sets correspond to the stable intersection of two Cantor sets in the plane.

Endoscopy, theta-liftings, and period integrals for the unitary group in three variables

Abstract: L-packets for the quasi-split unitary group in three variables U(3). We shall give an explicit parametrization of these L-packets using theta liftings and describe some relations between the structure of L-packets and certain period integrals. We also prove that an endoscopic L-packet contains a unique generic representation in local and global cases. Suppose that G is a reductive algebraic group over a local or global field

Central extensions of word hyperbolic groups

Abstract: perbolic groups by finitely generated abelian groups are automatic. We show that they are in fact biautomatic. Further, we show that every 2-dimensional cohomology class on a word hyperbolic group can be represented by a bounded 2-cocycle. This lends weight to the claim of Gromov that for a word hyperbolic group, the cohomology in every dimension > 2 is bounded. Our results apply more generally to virtually central extensions. We build on the ideas presented in [4], where the general problem was reduced to the case of central extensions by Z and was solved for Fuchsian groups. Some special cases of automaticity or biautomaticity in this case had previously been proved in [1], [2], and [7]. The new ingredient is a maximising technique inspired by work of Epstein and Fujiwara. Beginning with an arbitrary finite generating set for a central extension by 2, this maximising process is used to obtain a section which, in the language of [4], corresponds to a "regular 2-cocycle" on the hyperbolic group G, and can be used to obtain a biautomatic structure for the extension. Since central extensions correspond to 2-dimensional cohomology classes, it follows that every such class can be represented by a regular 2-cocycle. Using the geometric properties of G, we then further modify this cocycle to obtain a bounded representative for the original cohomology class. We also discuss the relations between various concepts of "weak boundedness" of a 2-cocycle on an arbitrary finitely generated group, related to quasi-isometry properties of central extensions. For cohomology classes, these weak boundedness concepts are shown to be equivalent to each other. We do not know if a weakly bounded cohomology class must be bounded.