Showing papers in "Journal of the American Mathematical Society in 2002"

Families of rationally connected varieties

Abstract: Recall that a proper variety X is said to be rationally connected if two general points p, q ∈ X are contained in the image of a map g : P → X. This is clearly a birationally invariant property. When X is smooth, this turns out to be equivalent to the a priori weaker condition that two general points can be joined by a chain of rational curves and also to the a priori stronger condition that for any finite subset Γ ⊂ X, there is a map g : P → X whose image contains Γ and such that gTX is an ample bundle.

Noetherian hereditary abelian categories satisfying Serre duality

Abstract: Norwegian Univ Sci & Technol, Dept Math Sci, N-7491 Trondheim, Norway. Limburgs Univ Ctr, Dept WNI, B-3590 Diepenbeek, Belgium.Reiten, I, Norwegian Univ Sci & Technol, Dept Math Sci, N-7491 Trondheim, Norway.

The method of alternating projections and the method of subspace corrections in Hilbert space

Abstract: The method of alternating projections and the method of subspace corrections are general iterative methods that have a variety of applications. The method of alternating projections, first proposed by von Neumann (1933) (see [31]), is an algorithm for finding the best approximation to any given point in a Hilbert space from the intersection of a finite number of subspaces. The method of subspace corrections, an abstraction of general linear iterative methods such as multigrid and domain decomposition methods, is an algorithm for finding the solution of a linear system of equations. In this paper, we shall study these two methods in a Hilbert space setting and in particular present a new identity for the product of nonexpansive operators that gives a sharpest possible estimate of the convergence rate of these methods. Let V be a Hilbert space and Vi ⊂ V (i = 1, . . . , J) a number of closed subspaces satisfying V = ∑J i=1 Vi. One main result in this paper is that the following identity holds for an appropriate class of operators Ti : V 7→ Vi (see Theorem 4.2 below):

Torification and factorization of birational maps

Abstract: Building on the work of the fourth author in math.AG/9904074, we prove the weak factorization conjecture for birational maps in characteristic zero: a birational map between complete nonsingular varieties over an algebraically closed field K of characteristic zero is a composite of blowings up and blowings down with smooth centers. Such a factorization exists which is functorial with respect to absolute isomorphisms, and compatible with a normal crossings divisor. The same holds for algebraic and analytic spaces. Another proof of the main theorem by the fourth author appeared in math.AG/9904076.

The nature of singularities in mean curvature flow of mean-convex sets

Abstract: Let K be a compact subset of R, or, more generally, of an (n+1)-dimensional riemannian manifold. We suppose that K is mean-convex. If the boundary of K is smooth and connected, this means that the mean curvature of ∂K is everywhere nonnegative (with respect to the inward unit normal) and is not identically 0. More generally, it means that Ft(K) is contained in the interior of K for t > 0, where Ft(K) is the set obtained by letting K evolve for time t under the level set mean curvature flow. As K evolves, it traces out a closed set K of spacetime: K = {(x, t) ∈ R ×R : x ∈ Ft(K)}. Also, there is associated to K a Brakke flow M : t 7→Mt of rectifiable varifolds. We call the pair (M,K) a mean-convex flow. Let X = (x, t) be a point in spacetime with t > 0. Suppose (xi, ti) is a sequence of points converging to X and λi is a sequence of positive numbers tending to infinity. Translate the pair M and K in spacetime by (y, τ) 7→ (y − xi, τ − ti) and then dilate parabolically by (y, τ) 7→ (λiy, λi τ) to get new flows Mi and Ki. The sequence (Mi,Ki) is called a blow-up sequence at X . General compactness theorems guarantee that this sequence will have subsequential limits. A subsequential limit (M′,K′) is called a limit flow. Here M′ : t ∈ (−∞,∞) 7→M ′ t

Blow up in finite time and dynamics of blow up solutions for the ²–critical generalized KdV equation

Abstract: In this paper, we are interested in the phenomenon of blow up in finite time (or formation of singularity in finite time) of solutions of the critical generalized KdV equation. Few results are known in the context of partial differential equations with a Hamiltonian structure. For the semilinear wave equation, or more generally for hyperbolic systems, the finite speed of propagation allows one to build blowing up solutions by reducing the problem to an ordinary differential equation. For the nonlinear Schrödinger equation, iut = −∆u− |u|p−1u, where u : R×R → C, (1)

Singularities of pairs via jet schemes

Abstract: Let X be a smooth variety and Y a closed subscheme of X. By comparing motivic integrals on X and on a log resolution of (X,Y), we prove the following formula for the log canonical threshold of (X,Y): c(X,Y)=dim X-sup_m{(dim Y_m}/(m+1)}, where Y_m is the mth jet scheme of Y. We show how this formula can be used to study the log canonical threshold. In particular, we give a proof of the Semicontinuity theorem of Demailly and Koll\'ar.

Classification of limiting shapes for isotropic curve flows

Abstract: with a 5 0, and initial condition x(p, 0) = xzo(p). This produces a family of curves yt = x(Si, t). Here n is the curvature, and n is the outward-pointing unit normal vector. These equations are particularly natural in that they are isotropic (equivariant under rotations in the plane) and homogeneous (equivariant under dilation of space, if time is also scaled accordingly). The main aim of this paper is to provide a complete description of the behaviour of embedded convex curves moving by equations of the form (1.1). In certain cases this description has already been provided: If a = 1 then Equation (1.1) is the curve-shortening flow, for which the following holds:

Regularity on abelian varieties I

Abstract: We introduce the notion of Mukai regularity ($M$-regularity) for coherent sheaves on abelian varieties. The definition is based on the Fourier-Mukai transform, and in a special case depending on the choice of a polarization it parallels and strengthens the usual Castelnuovo-Mumford regularity. Mukai regularity has a large number of applications, ranging from basic properties of linear series on abelian varieties and defining equations for their subvarieties, to higher dimensional type statements and to a study of special classes of vector bundles. Some of these applications are explained here, while others are the subject of upcoming sequels.

Construction of discrete series for classical p-adic groups

Abstract: In this paper the authors complete the classification of irreducible square integrable representations of classical $p$-adic groups, assuming a natural assumption which is expected to hold in general. This classification implies a parameterization of irreducible tempered representations of these groups and it implies a classification of the non-unitary duals of these groups (modulo cuspidal data).

Growth of solutions for QG and 2D Euler equations

Abstract: The work of Constantin-Majda-Tabak [1] developed an analogy between the Quasi-geostrophic and 3D Euler equations. Constantin, Majda and Tabak proposed a candidate for a singularity for the Quasi-geostrophic equation. Their numerics showed evidence of a blow-up for a particular initial data, where the level sets of the temperature contain a hyperbolic saddle. The arms of the saddle tend to close in finite time, producing a sharp front. Numerics studies done later by Ohkitani-Yamada [8] and Constantin-Nie-Schorghofer [2], with the same initial data, suggested that instead of a singularity the derivatives of the temperature were increasing as double exponential in time. The study of collapse on a curve was first studied in [1] for the Quasi-geostrophic equation where they considered a simplified ansatz for classical frontogenesis with trivial topology. At the time of collapse, the scalar θ is discontinuous across the curve x2 = f(x1) with different limiting values for the temperature on each side of the front. They show that under this topology the directional field remains smooth up to the collapse, which contradicts the following theorem proven in [1]:

Analytic continuation of overconvergent eigenforms

Abstract: Let f be an overconvergent p-adic eigenform of level Np r , r 1, with non-zero Up-eigenvalue. We show how f may be analytically continued to a subset of X1(Np r ) an containing, for example, all the supersingular locus. Using these results we extend the main theorem of [BT] to many ramified cases.

Totally positive Toeplitz matrices and quantum cohomology of partial flag varieties

Abstract: We show that the set of totally positive unipotent lower-triangular Toeplitz matrices in GLn form a real semi-algebraic cell of dimension n 1. Furthermore we prove a natural cell decomposition for its closure. The proof uses properties of the quantum cohomology rings of the partial flag varieties of GLn(C) relying in particular on the positivity of the structure constants, which are enumerative Gromov-Witten invariants. We also give a characterization of total positivity for Toeplitz matrices in terms of the (quantum) Schubert classes. This work builds on some results of Dale Peterson's which we explain with proofs in the type A case.

The classification problem for torsion-free abelian groups of finite rank

Abstract: In 1937, Baer [5] introduced the notion of the type of an element in a torsion-free abelian group and showed that this notion provided a complete invariant for the classification problem for torsion-free abelian groups of rank 1. Since then, despite the efforts of such mathematicians as Kurosh [23] and Malcev [25], no satisfactory system of complete invariants has been found for the torsion-free abelian groups of finite rank n > 2. So it is natural to ask whether the classification problem is genuinely more difficult for the groups of rank n > 2. Of course, if we wish to show that the classification problem for the groups of rank n > 2 is intractible, it is not enough merely to prove that there are 2' such groups up to isomorphism. For there are 2' pairwise nonisomorphic groups of rank 1, and we have already pointed out that Baer has given a satisfactory classification for this class of groups. In this paper, following Friedman-Stanley [11] and Hjorth-Kechris [15], we shall use the more sensitive notions of descriptive set theory to measure the complexity of the classification problem for the groups of rank n > 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space Qf which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank 1 < r < n can be naturally identified with the set S(Qn) of all nontrivial additive subgroups of Qn. Notice that S(Qn) is a Borel subset of the Polish space ,p(Qn) of all subsets of Qn, and hence S(Qn) can be regarded as a standard Borel space; i.e. a Polish space equipped with its associated o-algebra of Borel subsets. (Here we are identifying p(Qn) with the space 2'Qn of all functions h: Qn {0, 1} equipped with the product topology.) Furthermore, the natural action of GLn(Q) on the vector space Qn induces a corresponding Borel action on S(Qn); and it is easily checked that if A, B C S(Qn), then A _V B iff there exists an element f E GLn(Q) such that p(A) = B. It follows that the isomorphism relation on S(Qn) is a countable Borel equivalence relation. (If X is a standard Borel space, then a Borel equivalence relation on X is an equivalence relation E C X2 which is a Borel subset of X2. The Borel equivalence relation E is said to be countable iff every E-equivalence class is countable.)

Random polynomials having few or no real zeros

Abstract: Consider a polynomial of large degree n whose coefficients are independent, identically distributed, nondegenerate random variables having zero mean and finite moments of all orders. We show that such a polynomial has exactly k real zeros with probability n−b+o(1) as n → ∞ through integers of the same parity as the fixed integer k ≥ 0. In particular, the probability that a random polynomial of large even degree n has no real zeros is n−b+o(1). The finite, positive constant b is characterized via the centered, stationary Gaussian process of correlation function sech(t/2). The value of b depends neither on k nor upon the specific law of the coefficients. Under an extra smoothness assumption about the law of the coefficients, with probability n−b+o(1) one may specify also the approximate locations of the k zeros on the real line. The constant b is replaced by b/2 in case the i.i.d. coefficients have a nonzero mean.

Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable

Abstract: We prove that finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. Let D be a finite dimensional division algebra having center K and let N ⊆ D× be a normal subgroup of finite index. Suppose D×/N is not solvable. Then we may assume that H := D×/N is a minimal nonsolvable group (MNS group for short), i.e., a nonsolvable group all of whose proper quotients are solvable. Our proof now has two main ingredients. One ingredient is to show that the commuting graph of a finite MNS group satisfies a certain property which we denote property (3 12 ). This property includes the requirement that the diameter of the commuting graph should be ≥ 3, but is, in fact, stronger. Another ingredient is to show that if the commuting graph of D×/N has the property (312 ), then N is open with respect to a nontrivial height one valuation of D (assuming without loss, as we may, that K is finitely generated). After establishing the openness of N (when D×/N is an MNS group) we apply the Nonexistence Theorem whose proof uses induction on the transcendence degree of K over its prime subfield, to eliminate H as a possible quotient of D×, thereby obtaining a contradiction and proving our main result.

The planar Cantor sets of zero analytic capacity and the local ()-Theorem

Abstract: In this paper we obtain rather precise estimates for the analytic capacity of a big class of planar Cantors sets. In fact, we show that analytic capacity and positive analytic capacity are comparable for these sets. The main tool for the proof is an appropriate version of the T (b)-Theorem.

Supersingular elliptic curves, theta series and weight two modular forms

Abstract: This paper deals with two subjects and their interaction. The first is the problem of spanning spaces of modular forms by theta series. The second is the commutative algebraic properties of Hecke modules arising in the arithmetic theory of modular forms. Let p be a prime, and let B denote the quaternion algebra over Q that is ramified at p and ∞ and at no other places. If L is a left ideal in a maximal order of B then L is a rank four Z-module equipped in a natural way with a positive definite quadratic form [6, §1]. (We shall say that L is a rank four quadratic space, and remark that the isomorphism class of L as a quadratic space depends only on the left ideal class of L in its maximal order.) Eichler [5] proved that the theta series of L is a weight two modular form on Γ0(p), and that as L ranges over a collection of left ideal class representatives of all left ideals in all maximal orders of B these theta series span the vector space of weight two modular forms on Γ0(p) over Q. In this paper we strengthen this result as follows: if L is as above, then the q-expansion of its theta series Θ(L) has constant term equal to one and all other coefficients equal to even integers. Suppose that f is a modular form whose qexpansion coefficients are even integers, except perhaps for its constant term, which we require merely to be an integer. It follows from Eichler’s theorem that f may be written as a linear combination of Θ(L) (with L ranging over a collection of left ideals of maximal orders of B) with rational coefficients. We show that in fact these coefficients can be taken to be integers. Let T denote the Z-algebra of Hecke operators acting on the space of weight two modular forms on Γ0(p). The proof that we give of our result hinges on analyzing the structure of a certain T-module X . We can say what X is: it is the free Zmodule of divisors supported on the set of singular points of the (reducible, nodal) curve X0(p) in characteristic p. The key properties of X , which imply the above result on theta series, are that the natural map T −→ EndT(X ) is an isomorphism, and that furthermore X is locally free of rank one in a Zariski neighbourhood of the Eisenstein ideal of T. We remark that it is comparatively easy to prove the analogous statements after tensoring with Q, for they then follow from the fact that X is a faithful T-module. Indeed, combining this with the semi-simplicity of the Q-algebra T⊗Z Q, one deduces that X ⊗Z Q is a free T⊗Z Q-module of rank one, and in particular that the map T⊗Z Q −→ EndT⊗ZQ(X ⊗Z Q) is an isomorphism.

Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles

Abstract: paper is to characterize the rational cohomology ring of 7-n when the rank is 2 and the degree is odd. In fact, we have given a complete set of generators for this ring in another paper [14]. So it is now a question of finding the relations between these generators. Even though this is a natural companion to the generation problem, the ideas and methods with which it is studied have quite a different flavor. In particular, there is much more explicit calculation. What makes the rank 2 case tractable is that the number of generators is man-

Simple homogeneous models

Abstract: This paper attempts to give a self-contained development of dividing theory (also called forking theory) in a strongly homogeneous structure. Dividing is a combinatorial property of the invariant relations on a structure that have yielded deep results for the models of so-called "simple" first-order theories. Below we describe for the nonspecialist how this paper fits in the broader context of geometrical stability theory. Naturally, some background in first-order model theory helps to understand these motivating results; however virtually no knowledge of logic is assumed in this paper. Readers desiring a more thorough description of geometrical stability theory are referred to the surveys [Hru97] and [Hru98]. Traditionally, geometrical stability theory is a collection of results that apply to definable relations on arbitrary models of a complete first-order theory. It is equivalent and convenient to restrict our attention to the definable relations on a fixed representative model of the theory, called a universal domain. Using the terminology of the abstract, a universal domain is an uncountable model M equipped with the first-order definable relations R which is strongly IMI-homogeneous and compact; i.e., if {Xi: i E I}, where III < IMI, is a family of definable relations on M so that niEF Xi + 0 for any finite F C I, then niEI Xi + 0. For our purposes the reader can assume there is a one-to-one correspondence between (complete first-order) theories and universal domains. A massive amount of abstract model theory was developed en route to Shelah's proof of Morley's Conjecture about the number of models, ranging over uncountable cardinals, of a fixed first-order theory [She9O]. Most of the work concerned the case of a stable theory, which will not be defined here for the sake of brevity. What is relevant is that most theorems describing the models of a stable theory rely on the forking independence relation. The forking independence relation, F, is a ternary relation on the subsets of the universal domain of a theory, where F(A, B; C) is read "A is forking independent from B over C" (see Remark 2.2). In a stable theory forking independence is symmetric (over C), has finite character (in A and B), bounded dividing, the free extension property and is transitive. (See Definition 2.5, Theorem 2.14 and Corollary 2.15 for precise statements of these properties.) These properties facilitate the introduction of several notions of dimension that lead to procedures for determining when two models are isomorphic. The combinatorial-geometric properties of the definable relations reflected in these dimensions profoundly impact the structure of the models beyond the question of fixing an isomorphism type. The results connected to algebra, known as geometrical

Derived Hilbert schemes

Abstract: We construct the derived version of the Hilbert scheme parametrizing subschemes in a given projective scheme X with given Hilbert polynomial h. This is a dg-manifold (smooth dg-scheme) RHilb_h(X) which carries a natural family of commutative (up to homotopy) dg-algebras, which over the usual Hilbert scheme is just given by truncations of the homogeneous coordinate rings of subschemes in X. In particular, RHilb_h(X) differs from RQuot_h(O_X), the derived Quot scheme constructed in our previous paper (math.AG/9905174) which carries only a family of A-infinity modules over the coordinate algebra of X. As an application, we construct the derived version of the moduli stack of stable maps of (variable) algebraic curves to a given projective variety Y, thus realizing the original suggestion of M. Kontsevich.

The grafting map of Teichmüller space

Abstract: One of the underlying principles in the study of Kleinian groups is that aspects of the complex projective geometry of quotients of Ĉ by these groups reflect properties of the three-dimensional hyperbolic geometry of the quotients ofH by these groups. Yet, even though it has been over thirty-five years since Lipman Bers exhibited a holomorphic embedding of the Teichmuller space of Riemann surfaces in terms of the projective geometry of a Teichmuller space of quasi-Fuchsian manifolds, no corresponding embedding in terms of the three-dimensional hyperbolic geometry has been presented. One of the goals of this paper is to give such an embedding. This embedding is straightforward and has been expected for some time ([Ta97], [Mc98]): to each member of a Bers slice of the space QF of quasi-Fuchsian 3manifolds, we associate the bending measured lamination of the convex hull facing the fixed “conformal” end. The geometric relationship between a boundary component of a convex hull and the projective surface at infinity for its end is given by a process known as grafting, an operation on projective structures on surfaces that traces its roots back at least to Klein [Kl33, p. 230, §50], with a modern history developed by many authors ([Ma69], [He75], [Fa83], [ST83], [Go87], [GKM00], [Ta97], [Mc98]). The main technical tool in our proof that bending measures give coordinates for Bers slices, and the second major goal of this paper, is the completion of the proof of the “Grafting Conjecture”. This conjecture states that for a fixed measured lamination λ, the self-map of Teichmuller space induced by grafting a surface along λ is a homeomorphism of Teichmuller space; our contribution to this argument is a proof of the injectivity of the grafting map. While the principal application of this result that we give is to geometric coordinates on the Bers slice of QF , one expects that the grafting homeomorphism might lead to other systems of geometric coordinates for other families of Kleinian groups (see §5.2); thus we feel that this result is of interest in its own right. We now state our results and methods more precisely. Throughout, S will denote a fixed differentiable surface which is closed, orientable, and of genus g ≥ 2. Let Tg be the corresponding Teichmuller space of marked conformal structures on S, and let Pg denote the deformation space of (complex) projective structures on S (see §2 for definitions).

Obstructions to nonnegative curvature and rational homotopy theory

Abstract: We establish a link between rational homotopy theory and the problem which vector bundles admit a complete Riemannian metric of nonnegative sectional curvature. As an application, we show for a large class of simply-connected nonnegatively curved manifolds that, if C lies in the class and T is a torus of positive dimension, then "most" vector bundles over C x T admit no complete nonnegatively curved metrics.

General elephants of three-fold divisorial contractions

Abstract: The theory of minimal models has enriched the study of higher-dimensional algebraic geometry; see [10] and [12]. For a variety with mild singularities, this theory produces another variety which possesses good properties, after finite elementary transformations called divisorial contractions and flips. Since Mori completed this program in dimension three in [17], it has become desirable to study three-folds explicitly. This paper aims to complete the explicit study of three-fold divisorial contractions whose exceptional divisors contract to Gorenstein points, after the papers [7] and [8]. Reid pointed out that general elements in the anti-canonical systems of threefolds have at worst Du Val singularities in appropriate situations involving contractions of extremal faces. This has become known as the general elephant conjecture. The papers [19], [21] and [22] support it with affirmative answers for Fano threefolds with singularities, and this approach has settled the problem of the existence of three-fold flips in [11] and [17]. Our main theorem is that this conjecture holds for our divisorial contractions:

On a lattice problem of H. Steinhaus

Abstract: Sometime in the 1950's, Steinhaus posed the following problem. Do there exist two sets A and S in the plane such that every set congruent to A has exactly one point in common with S? The trivial case where one of the sets is the plane and the other consists of a single point is ruled out. The first appearance of this problem in the literature seems to be in a 1958 paper of Sierpin'ski [14]. In this paper, he showed the answer is yes, a result later rediscovered by Erdos [5]. Of course, there are many variants of this problem. For example, one could specify the set A. In this direction, Komjaith showed that such a set exists if A = Z, the set of all integers [13]. Steinhaus also asked about the specific case where A = 2. The first reference to this problem also seems to be Sierpin'ski's 1958 paper where he mentions that in this case there is no set S which is bounded and open or else bounded and closed. This specific problem has been widely noted (see e.g. [3, 4]), but has remained unsolved until now. In this paper we answer this question in the affirmative:

Measures induced by analytic functions and a problem of Walter Rudin

Abstract: Let H?? denote the set of bounded analytic functions in the unit disk U={ZEC: IzI