Showing papers in "Inventiones Mathematicae in 2013"

Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)

Abstract: In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations This framework, described in Sect 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose’s b-structures These include asymptotically de Sitter-type metrics on a blow-up of the natural compactification, Kerr-de Sitter-type metrics, as well as asymptotically Minkowski metrics The simplest application is a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces), including a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane These results are also available in a follow-up paper which is more expository in nature (Vasy in Uhlmann, G (ed) Inverse Problems and Applications Inside Out II, 2012) The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here

Kac’s program in kinetic theory

Abstract: This paper is devoted to the study of propagation of chaos and mean-field limits for systems of indistinguishable particles, undergoing collision processes. The prime examples we will consider are the many-particle jump processes of Kac and McKean (Kac in Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III, pp. 171–197, 1956; McKean in J. Comb. Theory 2:358–382, 1967) giving rise to the Boltzmann equation. We solve the conjecture raised by Kac (Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954–1955, Vol. III, pp. 171–197, 1956), motivating his program, on the rigorous connection between the long-time behavior of a collisional many-particle system and the one of its mean-field limit, for bounded as well as unbounded collision rates.

Suita conjecture and the Ohsawa-Takegoshi extension theorem

Abstract: We prove a conjecture of N. Suita which says that for any bounded domain D in ℂ one has $c_{D}^{2}\leq\pi K_{D}$ , where c D (z) is the logarithmic capacity of ℂ∖D with respect to z∈D and K D the Bergman kernel on the diagonal. We also obtain optimal constant in the Ohsawa-Takegoshi extension theorem.

Existence of log canonical closures

Abstract: Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U 0⊂U, such that (X,Δ)× U U 0 has a good minimal model over U 0 and the images of all the non-klt centers intersect U 0, then (X,Δ) has a good minimal model over U. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

Dissipative continuous Euler flows

Abstract: We show the existence of continuous periodic solutions of the 3D incompressible Euler equations which dissipate the total kinetic energy.

The Local Langlands Correspondence for GLn over p-adic fields

Abstract: We extend our methods from Scholze (Invent. Math. 2012, doi: 10.1007/s00222-012-0419-y ) to reprove the Local Langlands Correspondence for GL n over p-adic fields as well as the existence of l-adic Galois representations attached to (most) regular algebraic conjugate self-dual cuspidal automorphic representations, for which we prove a local-global compatibility statement as in the book of Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001). In contrast to the proofs of the Local Langlands Correspondence given by Henniart (Invent. Math. 139(2), 439–455, 2000), and Harris-Taylor (The Geometry and Cohomology of Some Simple Shimura Varieties, 2001), our proof completely by-passes the numerical Local Langlands Correspondence of Henniart (Ann. Sci. Ec. Norm. Super. 21(4), 497–544, 1988). Instead, we make use of a previous result from Scholze (Invent. Math. 2012, doi: 10.1007/s00222-012-0419-y ) describing the inertia-invariant nearby cycles in certain regular situations.

Motivic degree zero Donaldson-Thomas invariants

Abstract: Given a smooth complex threefold X, we define the virtual motive \([\operatorname{Hilb}^{n}(X)]_{\operatorname {vir}}\) of the Hilbert scheme of n points on X. In the case when X is Calabi–Yau, \([\operatorname{Hilb}^{n}(X)]_{\operatorname{vir}}\) gives a motivic refinement of the n-point degree zero Donaldson–Thomas invariant of X. The key example is X=ℂ3, where the Hilbert scheme can be expressed as the critical locus of a regular function on a smooth variety, and its virtual motive is defined in terms of the Denef–Loeser motivic nearby fiber. A crucial technical result asserts that if a function is equivariant with respect to a suitable torus action, its motivic nearby fiber is simply given by the motivic class of a general fiber. This allows us to compute the generating function of the virtual motives \([\operatorname{Hilb}^{n} (\mathbb{C}^{3})]_{\operatorname{vir}}\) via a direct computation involving the motivic class of the commuting variety. We then give a formula for the generating function for arbitrary X as a motivic exponential, generalizing known results in lower dimensions. The weight polynomial specialization leads to a product formula in terms of deformed MacMahon functions, analogous to Gottsche’s formula for the Poincare polynomials of the Hilbert schemes of points on surfaces.

Local models of Shimura varieties and a conjecture of Kottwitz

Abstract: We give a group theoretic definition of “local models” as sought after in the theory of Shimura varieties. These are projective schemes over the integers of a p-adic local field that are expected to model the singularities of integral models of Shimura varieties with parahoric level structure. Our local models are certain mixed characteristic degenerations of Grassmannian varieties; they are obtained by extending constructions of Beilinson, Drinfeld, Gaitsgory and the second-named author to mixed characteristics and to the case of general (tamely ramified) reductive groups. We study the singularities of local models and hence also of the corresponding integral models of Shimura varieties. In particular, we study the monodromy (inertia) action and show a commutativity property for the sheaves of nearby cycles. As a result, we prove a conjecture of Kottwitz which asserts that the semi-simple trace of Frobenius on the nearby cycles gives a function which is central in the parahoric Hecke algebra.

On the Davenport-Heilbronn theorems and second order terms

Abstract: We give simple proofs of the Davenport–Heilbronn theorems, which provide the main terms in the asymptotics for the number of cubic fields having bounded discriminant and for the number of 3-torsion elements in the class groups of quadratic fields having bounded discriminant. We also establish second main terms for these theorems, thus proving a conjecture of Roberts. Our arguments provide natural interpretations for the various constants appearing in these theorems in terms of local masses of cubic rings.

Rotational symmetry of self-similar solutions to the Ricci flow

Abstract: Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman’s first paper.

Tensor tomography on surfaces

Abstract: We show that on simple surfaces the geodesic ray transform acting on solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long standing inverse problem in the two-dimensional case.

Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem

Abstract: We consider the energy critical Schrodinger map problem with the 2-sphere target for equivariant initial data of homotopy index k=1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

Lower bounds on Ricci curvature and quantitative behavior of singular sets

Abstract: Let Yn denote the Gromov-Hausdorff limit \(M^{n}_{i}\stackrel{d_{\mathrm{GH}}}{\longrightarrow} Y^{n}\) of v-noncollapsed Riemannian manifolds with \({\mathrm{Ric}}_{M^{n}_{i}}\geq-(n-1)\). The singular set \(\mathcal {S}\subset Y\) has a stratification \(\mathcal {S}^{0}\subset \mathcal {S}^{1}\subset\cdots\subset \mathcal {S}\), where \(y\in \mathcal {S}^{k}\) if no tangent cone at y splits off a factor ℝk+1 isometrically. Here, we define for all η>0, 0

Weak and strong fillability of higher dimensional contact manifolds

Abstract: For contact manifolds in dimension three, the notions of weak and strong symplectic fillability and tightness are all known to be inequivalent. We extend these facts to higher dimensions: in particular, we define a natural generalization of weak fillings and prove that it is indeed weaker (at least in dimension five), while also being obstructed by all known manifestations of “overtwistedness”. We also find the first examples of contact manifolds in all dimensions that are not symplectically fillable but also cannot be called overtwisted in any reasonable sense. These depend on a higher dimensional analogue of Giroux torsion, which we define via the existence in all dimensions of exact symplectic manifolds with disconnected contact boundary.

On the \(\partial\overline{\partial}\)-Lemma and Bott-Chern cohomology

Abstract: On a compact complex manifold X, we prove a Frolicher-type inequality for Bott-Chern cohomology and we show that the equality holds if and only if X satisfies the \(\partial\overline{\partial}\)-Lemma.

Global existence of weak solutions to the FENE dumbbell model of polymeric flows

Abstract: Systems coupling fluids and polymers are of great interest in many branches of sciences. One of the most classical models to describe them is the FENE (Finite Extensible Nonlinear Elastic) dumbbell model. We prove global existence of weak solutions to the FENE dumbbell model of polymeric flows. The main difficulty is the passage to the limit in a nonlinear term that has no obvious compactness properties. The proof uses many weak convergence techniques. In particular it is based on the control of the propagation of strong convergence of some well chosen quantity by studying a transport equation for its defect measure. In addition, this quantity controls a rescaled defect measure of the gradient of the velocity.

W2,1 regularity for solutions of the Monge-Ampère equation

Abstract: In this paper we prove that a strictly convex Alexandrov solution u of the Monge–Ampere equation, with right-hand side bounded away from zero and infinity, is $W^{2,1}_{\mathrm{loc}}$ . This is obtained by showing higher integrability a priori estimates for D 2 u, namely D 2 u∈Llog k L for any k∈ℕ.

Deformation limits of projective manifolds: Hodge numbers and strongly Gauduchon metrics

Abstract: This paper is intended to start a series of works aimed at proving that if in a (smooth) complex analytic family of compact complex manifolds all the fibres, except one, are supposed to be projective, then the remaining (limit) fibre must be Moishezon. A new method of attack, whose starting point originates in Demailly’s work, is introduced. While we hope to be able to address the general case in the near future, two important special cases are established here: the one where the Hodge numbers h0,1 of the fibres are supposed to be locally constant and the one where the limit fibre is assumed to be a strongly Gauduchon manifold. The latter is a rather weak metric assumption giving rise to a new, rather general, class of compact complex manifolds that we hereby introduce and whose relevance to this type of problems we underscore.

The Tate conjecture for K3 surfaces over finite fields

Abstract: Artin’s conjecture states that supersingular K3 surfaces over finite fields have Picard number 22. In this paper, we prove Artin’s conjecture over fields of characteristic p≥5. This implies Tate’s conjecture for K3 surfaces over finite fields of characteristic p≥5. Our results also yield the Tate conjecture for divisors on certain holomorphic symplectic varieties over finite fields, with some restrictions on the characteristic. As a consequence, we prove the Tate conjecture for cycles of codimension 2 on cubic fourfolds over finite fields of characteristic p≥5.

Unique isoperimetric foliations of asymptotically flat manifolds in all dimensions

Abstract: The question of isoperimetry What is the largest amount of volume that can be enclosed by a given amount of area? can be traced back to antiquity.1 The first mathematically rigorous results are as recent as the nineteenth century. The question of isoperimetry and its close relative, the analysis of minimal surfaces, are two of the model problems of the geometric calculus of variations. The list of geometries where an explicit answer to the question of isoperimetry is available is short. We provide an overview of available results in Appendix H. In this paper and [22], we extend this list by a class of Riemannian manifolds (M,g) for which we describe all large isoperimetric regions completely. We refer the reader to Sect. 2 for the precise definitions of all terms in the statement of our first main theorem:

The maximal number of exceptional Dehn surgeries

Abstract: Thurston’s hyperbolic Dehn surgery theorem is one of the most important results in 3-manifold theory, and it has stimulated an enormous amount of research. If M is a compact orientable hyperbolic 3-manifold with boundary a single torus, then the theorem asserts that, for all but finitely many slopes s on ∂M , the manifold M(s) obtained by Dehn filling along s also admits a hyperbolic structure. The slopes s where M(s) is not hyperbolic are known as exceptional. A major open question has been: what is the maximal number of exceptional slopes on such a manifold M? When M is the exterior of the figure-eight knot, the number of exceptional slopes is 10, and this was conjectured by Gordon in [19] to be an upper bound that holds for all M . In this paper, we prove this conjecture.

Spectral shift function of higher order

Abstract: This paper resolves affirmatively Koplienko’s (Sib. Mat. Zh. 25:62–71, 1984) conjecture on existence of higher order spectral shift measures. Moreover, the paper establishes absolute continuity of these measures and, thus, existence of the higher order spectral shift functions. A spectral shift function of order n∈ℕ is the function ηn=ηn,H,V such that $$ \operatorname {Tr}\Biggl( f(H + V)-\sum_{k = 0}^{n-1} \frac{1}{k!}\, \frac{d^k}{dt^k} \bigl[ f(H + tV) \bigr] \bigg|_{t = 0} \Biggr) = \int_\mathbb{R}f^{(n)} (t)\, \eta_n (t)\, dt, $$ for every sufficiently smooth function f, where H is a self-adjoint operator defined in a separable Hilbert space ℍ and V is a self-adjoint operator in the n-th Schatten-von Neumann ideal Sn. Existence and summability of η1 and η2 were established by Krein (Mat. Sb. 33:597–626, 1953) and Koplienko (Sib. Mat. Zh. 25:62–71, 1984), respectively, whereas for n>2 the problem was unresolved. We show that ηn,H,V exists, integrable, and $$\Vert \eta_n \Vert _{L^1(\mathbb{R})} \leq c_n \Vert V \Vert _{S^n}^n, $$ for some constant cn depending only on n∈ℕ. Our results for ηn rely on estimates for multiple operator integrals obtained in this paper. Our method also applies to the general semi-finite von Neumann algebra setting of the perturbation theory.

Cutoff for the Ising model on the lattice

Abstract: Introduced in 1963, Glauber dynamics is one of the most practiced and extensively studied methods for sampling the Ising model on lattices. It is well known that at high temperatures, the time it takes this chain to mix in L 1 on a system of size n is O(logn). Whether in this regime there is cutoff, i.e. a sharp transition in the L 1-convergence to equilibrium, is a fundamental open problem: If so, as conjectured by Peres, it would imply that mixing occurs abruptly at (c+o(1))logn for some fixed c>0, thus providing a rigorous stopping rule for this MCMC sampler. However, obtaining the precise asymptotics of the mixing and proving cutoff can be extremely challenging even for fairly simple Markov chains. Already for the one-dimensional Ising model, showing cutoff is a longstanding open problem. We settle the above by establishing cutoff and its location at the high temperature regime of the Ising model on the lattice with periodic boundary conditions. Our results hold for any dimension and at any temperature where there is strong spatial mixing: For ℤ2 this carries all the way to the critical temperature. Specifically, for fixed d≥1, the continuous-time Glauber dynamics for the Ising model on (ℤ/nℤ) d with periodic boundary conditions has cutoff at (d/2λ ∞)logn, where λ ∞ is the spectral gap of the dynamics on the infinite-volume lattice. To our knowledge, this is the first time where cutoff is shown for a Markov chain where even understanding its stationary distribution is limited. The proof hinges on a new technique for translating L 1-mixing to L 2-mixing of projections of the chain, which enables the application of logarithmic-Sobolev inequalities. The technique is general and carries to other monotone and anti-monotone spin-systems, e.g. gas hard-core, Potts, anti-ferromagentic Ising, arbitrary boundary conditions, etc.

Integrable measure equivalence and rigidity of hyperbolic lattices

Abstract: We study rigidity properties of lattices in \(\operatorname {Isom}(\mathbf {H}^{n})\simeq \mathrm {SO}_{n,1}({\mathbb{R}})\), n≥3, and of surface groups in \(\operatorname {Isom}(\mathbf {H}^{2})\simeq \mathrm {SL}_{2}({\mathbb{R}})\) in the context of integrable measure equivalence. The results for lattices in \(\operatorname {Isom}(\mathbf {H}^{n})\), n≥3, are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable measure equivalence classification. Despite the lack of Mostow rigidity for n=2 we show that cocompact lattices in \(\operatorname {Isom}(\mathbf {H}^{2})\) allow a similar integrable measure equivalence classification.

3-Manifolds with nonnegative Ricci curvature

Abstract: For a complete noncompact 3-manifold with nonnegative Ricci curvature, we prove that either it is diffeomorphic to ℝ3 or the universal cover splits. This confirms Milnor’s conjecture in dimension 3.

The index of an algebraic variety

Abstract: Let K be the field of fractions of a Henselian discrete valuation ring \({{\mathcal {O}}_{K}}\). Let XK/K be a smooth proper geometrically connected scheme admitting a regular model \(X/{{\mathcal {O}}_{K}}\). We show that the index δ(XK/K) of XK/K can be explicitly computed using data pertaining only to the special fiber Xk/k of the model X.

Bounds on the density of states for Schrödinger operators

Abstract: We establish bounds on the density of states measure for Schrodinger operators. These are deterministic results that do not require the existence of the density of states measure, or, equivalently, of the integrated density of states. The results are stated in terms of a “density of states outer-measure” that always exists, and provides an upper bound for the density of states measure when it exists. We prove log-Holder continuity for this density of states outer-measure in one, two, and three dimensions for Schrodinger operators, and in any dimension for discrete Schrodinger operators.

The main conjecture of Iwasawa theory for totally real fields

Abstract: Let p be an odd prime. Let $\mathcal{G}$ be a compact p-adic Lie group with a quotient isomorphic to ℤ p . We give an explicit description of K 1 of the Iwasawa algebra of $\mathcal{G}$ in terms of Iwasawa algebras of Abelian subquotients of $\mathcal{G}$ . We also prove a result about K 1 of a certain canonical localisation of the Iwasawa algebra of $\mathcal{G}$ , which occurs in the formulation of the main conjectures of noncommutative Iwasawa theory. These results predict new congruences between special values of Artin L-functions, which we then prove using the q-expansion principle of Deligne-Ribet. As a consequence we prove the noncommutative main conjecture for totally real fields, assuming a suitable version of Iwasawa’s conjecture about vanishing of the cyclotomic μ-invariant. In particular, we get an unconditional result for totally real pro-p p-adic Lie extension of Abelian extensions of ℚ.

The p -adic Gross-Zagier formula for elliptic curves at supersingular primes

Abstract: Let p be a prime number and let E be an elliptic curve defined over ℚ of conductor N. Let K be an imaginary quadratic field with discriminant prime to pN such that all prime factors of N split in K. B. Perrin-Riou established the p-adic Gross-Zagier formula that relates the first derivative of the p-adic L-function of E over K to the p-adic height of the Heegner point for K when E has good ordinary reduction at p. In this article, we prove the p-adic Gross-Zagier formula of E for the cyclotomic ℤ p -extension at good supersingular prime p. Our result has an application for the full Birch and Swinnerton-Dyer conjecture. Suppose that the analytic rank of E over ℚ is 1 and assume that the Iwasawa main conjecture is true for all good primes and the p-adic height pairing is not identically equal to zero for all good ordinary primes, then our result implies the full Birch and Swinnerton-Dyer conjecture up to bad primes. In particular, if E has complex multiplication and of analytic rank 1, the full Birch and Swinnerton-Dyer conjecture is true up to a power of bad primes and 2.

Applications of the Kuznetsov formula on GL(3)

Abstract: We develop a fairly explicit Kuznetsov formula on GL(3) and discuss the analytic behavior of the test functions on both sides. Applications to Weyl’s law, exceptional eigenvalues, a large sieve and L-functions are given.