Showing papers in "Inventiones Mathematicae in 2006"

Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case

Abstract: We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.

The trajectories of particles in Stokes waves

Abstract: Analyzing a free boundary problem for harmonic functions we show that there are no closed particle paths in an irrotational inviscid traveling wave propagating at the surface of water over a flat bed: within a period each particle experiences a backward-forward motion with a slight forward drift.

Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, II

Abstract: We consider crossed product II1 factors $M = N\rtimes_{\sigma}G$ , with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G on finite von Neumann algebras N that are “malleable” and mixing. Examples are the actions of G by Bernoulli shifts (classical and non-classical) and by Bogoliubov shifts. We prove a rigidity result for isomorphisms of such factors, showing the uniqueness, up to unitary conjugacy, of the position of the group von Neumann algebra L(G) inside M. We use this result to calculate the fundamental group of M, $\mathcal{F}(M)$ , in terms of the weights of the shift σ, for $G=\mathbb{Z}^2\rtimes SL(2,\mathbb{Z})$ and other special arithmetic groups. We deduce that for any subgroup S⊂ℝ+ * there exist II1 factors M (separable if S is countable or S=ℝ+ *) with $\mathcal{F}(M)=S$ . This brings new light to a long standing open problem of Murray and von Neumann.

Rigid modules over preprojective algebras

Abstract: Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis.

Anosov flows, surface groups and curves in projective space

Abstract: Note that in [10], W. Goldman gives a complete description of these connected components in the case of finite covers of PSL(2,R). In the case of PSL(2,R), two homeomorphic components, called Teichmuller spaces, play a central role. These two components are well known to be homeomorphic to a ball of dimension 6g − 6. N. Hitchin generalises this situation to PSL(n,R). Indeed, one of these components when n is odd, and two when n is even, has a very simple topology. Let us define an n-Fuchsian representation to be a representation ρ which can be written as ρ = ι ◦ ρ0, where ρ0 is a cocompact representation with values in PSL(2,R) and ι is the irreducible representation of PSL(2,R) in PSL(n,R), We denote by RepH(π1(S), PSL(n,R)) a connected component that contains Fuchsian representations, and call it a Hitchin component. In fact, when n is odd there is one Hitchin component, and when n is even two isomorphic ones. N. Hitchin proves in [17]

Murphy’s law in algebraic geometry: Badly-behaved deformation spaces

Abstract: We consider the question: “How bad can the deformation space of an object be?” The answer seems to be: “Unless there is some a priori reason otherwise, the deformation space may be as bad as possible.” We show this for a number of important moduli spaces. More precisely, every singularity of finite type over ℤ (up to smooth parameters) appears on: the Hilbert scheme of curves in projective space; and the moduli spaces of smooth projective general-type surfaces (or higher-dimensional varieties), plane curves with nodes and cusps, stable sheaves, isolated threefold singularities, and more. The objects themselves are not pathological, and are in fact as nice as can be: the curves are smooth, the surfaces are automorphism-free and have very ample canonical bundle, the stable sheaves are torsion-free of rank 1, the singularities are normal and Cohen-Macaulay, etc. This justifies Mumford’s philosophy that even moduli spaces of well-behaved objects should be arbitrarily bad unless there is an a priori reason otherwise. Thus one can construct a smooth curve in projective space whose deformation space has any given number of components, each with any given singularity type, with any given non-reduced behavior. Similarly one can give a surface over $\mathbb{F}_{p}$ that lifts to ℤ/p 7 but not ℤ/p 8. (Of course the results hold in the holomorphic category as well.) It is usually difficult to compute deformation spaces directly from obstruction theories. We circumvent this by relating them to more tractable deformation spaces via smooth morphisms. The essential starting point is Mnev’s universality theorem.

Boundedness of pluricanonical maps of varieties of general type

Abstract: Using the techniques of [20] and [10], we prove that certain log forms may be lifted from a divisor to the ambient variety. As a consequence of this result, following [22], we show that: For any positive integer n there exists an integer r n such that if X is a smooth projective variety of general type and dimension n, then $\phi_{rK_X}\colon X\dasharrow\mathbb{P}(H^0(\mathcal{O}_{X}(rK_X)))$ is birational for all r≥r n .

Mirror symmetry for Del Pezzo surfaces: Vanishing cycles and coherent sheaves

Abstract: We study homological mirror symmetry for Del Pezzo surfaces and their mirror Landau-Ginzburg models. In particular, we show that the derived category of coherent sheaves on a Del Pezzo surface Xk obtained by blowing up ℂℙ2 at k points is equivalent to the derived category of vanishing cycles of a certain elliptic fibration Wk:Mk→ℂ with k+3 singular fibers, equipped with a suitable symplectic form. Moreover, we also show that this mirror correspondence between derived categories can be extended to noncommutative deformations of Xk, and give an explicit correspondence between the deformation parameters for Xk and the cohomology class [B+iω]∈H2(Mk,ℂ).

The f(q) mock theta function conjecture and partition ranks

Abstract: In 1944, Freeman Dyson initiated the study of ranks of integer partitions. Here we solve the classical problem of obtaining formulas for N e (n) (resp. N o (n)), the number of partitions of n with even (resp. odd) rank. Thanks to Rademacher’s celebrated formula for the partition function, this problem is equivalent to that of obtaining a formula for the coefficients of the mock theta function f(q), a problem with its own long history dating to Ramanujan’s last letter to Hardy. Little was known about this problem until Dragonette in 1952 obtained asymptotic results. In 1966, G.E. Andrews refined Dragonette’s results, and conjectured an exact formula for the coefficients of f(q). By constructing a weak Maass-Poincare series whose “holomorphic part” is q -1 f(q 24), we prove the Andrews-Dragonette conjecture, and as a consequence obtain the desired formulas for N e (n) and N o (n).

The Weyl groupoid of a Nichols algebra of diagonal type

Abstract: The theory of Nichols algebras of diagonal type is known to be closely related to that of semi-simple Lie algebras. In this paper the connection between both theories is made closer. For any Nichols algebra of diagonal type invertible transformations are introduced, which remind one of the action of the Weyl group on the root system associated to a semi-simple Lie algebra. They give rise to the definition of a groupoid. As an application an alternative proof of classification results of Rosso, Andruskiewitsch, and Schneider is obtained without using any technical assumptions on the braiding.

Finiteness of relative equilibria of the four-body problem

Abstract: We show that the number of relative equilibria of the Newtonian four-body problem is finite, up to symmetry In fact, we show that this number is always between 32 and 8472 The proof is based on symbolic and exact integer computations which are carried out by computer

On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms

Abstract: The goal of this paper is to illustrate how the techniques of locally analytic p-adic representation theory (as developed in [28, 29, 30, 31] and [13, 14, 17]; see also [16] for a short summary of some of these results) may be applied to study arithmetic properties of automorphic representations. More specifically, we consider the problem of p-adically interpolating the systems of eigenvalues attached to automorphic Hecke eigenforms (as well as the corresponding Galois representations, in situations where these appear in the étale cohomology of Shimura varieties). We can summarize our approach to the problem as follows: rather than attempting to directly interpolate the systems of eigenvalues attached to eigenforms, we instead attempt to interpolate the automorphic representations that these eigenforms give rise to. To be more precise, we fix a connected reductive linear algebraic group G defined over a number field F , and a finite prime p of F . We let Fp denote the completion of F at p, let E be a finite extension of Fp over which the group G splits, let A denote the ring of adèles of F , and let Af denote the ring of finite adèles of F . The representations that we construct are admissible locally analytic representations of the group G(Af ) on certain locally convex topological E-vector spaces. These representations are typically not irreducible; rather, they contain as closed subrepresentations many locally algebraic representations of G(Af ) which are closely related to automorphic representations of G(A) of cohomological type. (It is for this reason that we regard the representations that we construct as forming an “interpolation” of those automorphic representations.) Once we have our locally analytic representations of G(Af ) in hand, we may apply to them the Jacquet module functors of [14]. In this way we obtain p-adic analytic families of systems of Hecke eigenvalues, which (under a suitable hypothesis, for which see the statement of Theorem 0.7 below) p-adically interpolate (in the

Moment analysis for localization in random Schrödinger operators

Abstract: We study localization effects of disorder on the spectral and dynamical properties of Schrodinger operators with random potentials. The new results include exponentially decaying bounds on the transition amplitude and related projection kernels, including in the mean. These are derived through the analysis of fractional moments of the resolvent, which are finite due to the resonance-diffusing effects of the disorder. The main difficulty which has up to now prevented an extension of this method to the continuum can be traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms. The difficulty is avoided here through the use of a weak-L 1 estimate concerning the boundary-value distribution of resolvents of maximally dissipative operators, combined with standard tools of relative compactness theory.

The subconvexity problem for Rankin-Selberg $L$-functions and equidistribution of Heegner points. II

Abstract: We prove a general subconvex bound in the level aspect for Rankin–Selberg L-functions associated with two primitive holomorphic or Maass cusp forms over Q. We use this bound to establish the equidistribution of incomplete Galois orbits of Heegner points on Shimura curves associated with indefinite quaternion algebras over Q.

Inversion of adjunction on log canonicity

Abstract: We prove inversion of adjunction on log canonicity.

Quadratic estimates and functional calculi of perturbed Dirac operators

Abstract: We prove quadratic estimates for complex perturbations of Dirac-type operators, and thereby show that such operators have a bounded functional calculus As an application we show that spectral projections of the Hodge–Dirac operator on compact manifolds depend analytically on L ∞ changes in the metric We also recover a unified proof of many results in the Calderon program, including the Kato square root problem and the boundedness of the Cauchy operator on Lipschitz curves and surfaces

Variation of Iwasawa invariants in Hida families

Abstract: Let ρ : GQ → GL2(k) be an absolutely irreducible modular Galois representation over a finite field k of characteristic p Assume further that ρ is p-ordinary and p-distinguished in the sense that the restriction of ρ to a decomposition group at p is reducible and non-scalar The Hida family H(ρ) of ρ is the set of all p-ordinary p-stabilized newforms f with mod p Galois representation isomorphic to ρ (If ρ is unramified at p, then one must also fix an unramified line in ρ and require that the ordinary line of f reduces to this fixed line) These newforms are a dense set of points in a certain p-adic analytic space of overconvergent eigenforms, consisting of an intersecting system of branches (ie irreducible components) T(a) indexed by the minimal primes a of a certain Hecke algebra To each modular form f ∈ H(ρ) one may associate the Iwasawa invariants μan( f ), λan( f ), μalg( f ) and λalg( f ) The analytic (resp algebraic) λ-invariants are the number of zeroes of the p-adic L-function (resp of the characteristic power series of the dual of the Selmer group) of f , while the μ-invariants are the exponents of the powers of p dividing the same objects In this paper we prove the following results on the behavior of these Iwasawa invariants as f varies over H(ρ)

Virasoro constraints for target curves

Abstract: We prove generalized Virasoro constraints for the relative Gromov-Witten theories of all nonsingular target curves. Descendents of the even cohomology classes are studied first by localization, degeneration, and completed cycle methods. Descendents of the odd cohomology are then controlled by monodromy and geometric vanishing relations. As an outcome of our results, the relative theories of target curves are completely and explicitly determined.

Finite covers of random 3-manifolds

Abstract: A 3-manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3-manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3-manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3-manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3-manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3-manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are led to consider the action of the mapping class group of a surface Σ on the set of quotients π1(Σ)→Q. If Q is a simple group, we show that if the genus of Σ is large, then this action is very mixing. In particular, the action factors through the alternating group of each orbit. This is analogous to Goldman’s theorem that the action of the mapping class group on the SU(2) character variety is ergodic.

Factor and normal subgroup theorems for lattices in products of groups

Abstract: A central result in the theory of semisimple groups and their lattices is Margulis’ normal subgroup theorem: any normal subgroup of an irreducible lattice in a center free, higher rank semisimple group, has finite index [Mar79,Mar91]. In the present paper we establish a Margulis-type theorem for a large family of lattices, including all Kac-Moody groups over (sufficiently large) finite fields. As in Margulis’ strategy, we establish along the way a “factor theorem” for measurable quotients of boundaries, which is of independent interest. Its proof introduces new ideas relying heavily on Furstenberg’s boundary theory (pertaining to harmonic functions, random walks and stationary measures for group actions). An adelic extension of the latter, and factor theorems in which the boundary is not a homogeneous space, follow as well. Recall that a group is called just infinite, if every non-trivial normal subgroup of it has finite index. The elementary observation that every finitely generated infinite group admits an infinite just infinite quotient, is one motivation for studying this property (see [Wil00] for more on the general structure of such groups). Extending this notion, we shall call a topological group G just non-compact, if every non-trivial closed normal subgroup N G is co-compact, and topologically just infinite, if every such N has finite index. Of course, for an abstract group G, all the three notions agree when it is viewed as a topological group with discrete topology.

Representation dimension of exterior algebras

Abstract: We determine the representation dimension of exterior algebras. This provides the first known examples of representation dimension > 3. We deduce that the Loewy length of the group algebra over F2 of a finite group is strictly bounded below by the 2-rank of the group (a conjecture of Benson). A key tool is the use of the concept of dimension of a triangulated category.

Singular symplectic moduli spaces

Abstract: Moduli spaces of semistable sheaves on a K3 or abelian surface with respect to a general ample divisor are shown to be locally factorial, with the exception of symmetric products of a K3 or abelian surface and the class of moduli spaces found by O’Grady. Consequently, since singular moduli space that do not belong to these exceptional cases have singularities in codimension ≥4 they do no admit projective symplectic resolutions.

Eigenvalue problem and a new product in cohomology of flag varieties

Abstract: Let G be a connected semisimple complex algebraic group and let P be a parabolic subgroup. In this paper we define a new (commutative and associative) product on the cohomology of the homogenous spaces G/P and use this to give a more efficient solution of the eigenvalue problem and also for the problem of determining the existence of G-invariants in the tensor product of irreducible representations of G. On the other hand, we show that this new product is intimately connected with the Lie algebra cohomology of the nil-radical of P via some works of Kostant and Kumar. We also initiate a uniform study of the geometric Horn problem for an arbitrary group $G$ by obtaining two (a priori) different sets of necessary recursive conditions to determine when a cohomology product of Schubert classes in G/P is non-zero. Hitherto, this was studied largely only for the group \SL(n).

The Monge-Ampère operator and geodesics in the space of Kähler potentials

Abstract: It is shown that geodesics in the space of K\\\"ahler potentials can be uniformly approximated by geodesics in the spaces of Bergman metrics. Two important tools in the proof are the Tian-Yau-Zelditch approximation theorem for K\\\"ahler potentials and the pluripotential theory of Bedford-Taylor, suitably adapted to K\\\"ahler manifolds.

Fibration de Hitchin et endoscopie

Abstract: We propose a geometric interpretation of the theory of elliptic endoscopy, due to Langlands and Kottwitz, in terms of the Hitchin fibration. As applications, we prove a global analog of a purity conjecture, due to Goresky, Kottwitz and MacPherson. For unitary groups, this global purity statement has been used, in a joint work with G. Laumon, to prove the fundamental lemma over a local fields of equal characteristics.

Nonsingular star flows satisfy Axiom A and the no-cycle condition

Abstract: We give an affirmative answer to a problem of Liao and Mane which asks whether, for a nonsingular flow to loose the Ω-stability, it must go through a critical-element-bifurcation. More precisely, a vector field S on a compact boundaryless manifold is called a star system if S has a C 1 neighborhood \(\mathcal{U}\) in the set of C 1 vector fields such that every singularity and every periodic orbit of every \(X\in\mathcal{U}\) is hyperbolic. We prove that any nonsingular star flow satisfies Axiom A and the no cycle condition.