Showing papers in "Duke Mathematical Journal in 2009"
TL;DR: The CLE(κ) ensembles as discussed by the authors are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties.
Abstract: We construct and study the conformal loop ensembles CLE(κ), defined for 8/3≤κ≤8, using branching variants of SLE(κ) called exploration trees. The CLE(κ) are random collections of countably many loops in a planar domain that are characterized by certain conformal invariance and Markov properties. We conjecture that they are the scaling limits of various random loop models from statistical physics, including the O(n) loop models
256 citations
TL;DR: In this article, the authors prove the existence of radial energy solutions of ∂ttu−Δu−u5=0 (0.1) in R3+1 which blow up exactly at r=t=0 as t→0−.
Abstract: Given ν>1/2 and δ>0 arbitrary, we prove the existence of energy solutions of ∂ttu−Δu−u5=0 (0.1) in R3+1 which blow up exactly at r=t=0 as t→0−. These solutions are radial and of the form u=λ(t)1/2W(λ(t)r)+η(r,t) inside the cone r≤t, where λ(t)=t−1−ν, W(r)=(1+r2/3)−1/2 is the stationary solution of (0.1), and η is a radiation term with ∫[r≤t](|∇η(x,t)|2+|ηt(x,t)|2+|η(x,t)|6)dx→0, t→0. Outside of the light-cone, there is the energy bound ∫[r>t](|∇u(x,t)|2+|ut(x,t)|2+|u(x,t)|6)dx 0. The regularity of u increases with ν. As in our accompanying article on wave maps [10], the argument is based on a renormalization method for the “soliton profile” W(r)
196 citations
TL;DR: In this article, the genus-zero twisted Gromov-Witten invariants of stable maps to a manifold or orbifold X were derived and applied to several examples.
Abstract: Twisted Gromov-Witten invariants are intersection numbers in moduli spaces of stable maps to a manifold or orbifold X which depend in addition on a vector bundle over X and an invertible multiplicative characteristic class. Special cases are closely related to local Gromov-Witten invariants of the bundle and to genus-zero one-point invariants of complete intersections in X. We develop tools for computing genus-zero twisted Gromov-Witten invariants of orbifolds and apply them to several examples. We prove a “quantum Lefschetz theorem” that expresses genus-zero one-point Gromov-Witten invariants of a complete intersection in terms of those of the ambient orbifold X. We determine the genus-zero Gromov-Witten potential of the type A surface singularity [C2/Zn]. We also compute some genus-zero invariants of [C3/Z3], verifying predictions of Aganagic, Bouchard, and Klemm [3]. In a self-contained appendix, we determine the relationship between the quantum cohomology of the An surface singularity and that of its crepant resolution, thereby proving the Crepant Resolution Conjectures of Ruan and of Bryan and Graber [12] in this case
178 citations
TL;DR: In this article, the authors define Floer homology for a time-independent or autonomous Hamiltonian on a symplectic manifold with contact-type boundary under the assumption that its 1-periodic orbits are transversally nondegenerate.
Abstract: We define Floer homology for a time-independent or autonomous Hamiltonian on a symplectic manifold with contact-type boundary under the assumption that its 1-periodic orbits are transversally nondegenerate. Our construction is based on Morse-Bott techniques for Floer trajectories. Our main motivation is to understand the relationship between the linearized contact homology of a fillable contact manifold and the symplectic homology of its filling
151 citations
TL;DR: The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole.
Abstract: The positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the Euclidean metric. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal hypersurface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole (see [HI]). In 1999, Bray extended this result to the general case of multiple black holes using a different technique (see [Br]). In this article, we extend the technique of [Br] to dimensions less than eight. Part of the argument is contained in a companion article by Lee [L]. The equality case of the theorem requires the added assumption that the manifold be spin
145 citations
TL;DR: In this paper, the authors elucidate the key role played by formality in the theory of characteristic and resonance varieties, and show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal.
Abstract: We elucidate the key role played by formality in the theory of characteristic and resonance varieties. We define relative characteristic and resonance varieties, V_k and R_k, related to twisted group cohomology with coefficients of arbitrary rank. We show that the germs at the origin of V_k and R_k are analytically isomorphic, if the group is 1-formal; in particular, the tangent cone to V_k at 1 equals R_k. These new obstructions to 1-formality lead to a striking rationality property of the usual resonance varieties. A detailed analysis of the irreducible components of the tangent cone at 1 to the first characteristic variety yields powerful obstructions to realizing a finitely presented group as the fundamental group of a smooth, complex quasi-projective algebraic variety. This sheds new light on a classical problem of J.-P. Serre. Applications to arrangements, configuration spaces, coproducts of groups, and Artin groups are given.
125 citations
TL;DR: In this article, the authors studied CMV matrices (discrete one-dimensional Dirac-type operators) with random decaying coefficients under mild assumptions, and identified the local eigenvalue statistics in the natural scaling limit.
Abstract: We study CMV matrices (discrete one-dimensional Dirac-type operators) with random decaying coefficients Under mild assumptions, we identify the local eigenvalue statistics in the natural scaling limit For rapidly decreasing coefficients, the eigenvalues have rigid spacing (like the numerals on a clock); in the case of slow decrease, the eigenvalues are distributed according to a Poisson process For a certain critical rate of decay, we obtain the β-ensembles of random matrix theory The temperature β−1 appears as the square of the coupling constant
108 citations
TL;DR: In this paper, a simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold is given.
Abstract: We give an efficient simplicial formula for the volume and Chern-Simons invariant of a boundary-parabolic PSL(2,C)-representation of a tame 3-manifold. If the representation is the geometric representation of a hyperbolic 3-manifold, our formula computes the volume and Chern-Simons invariant directly from an ideal triangulation with no use of additional combinatorial topology. In particular, the Chern-Simons invariant is computed just as easily as the volume
94 citations
TL;DR: The authors formulated a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe.
Abstract: We formulate a Serre-type conjecture for n-dimensional Galois representations that are tamely ramified at p. The weights are predicted using a representation-theoretic recipe. For n=3, some of these weights were not predicted by the previous conjecture of Ash, Doud, Pollack, and Sinnott. Computational evidence for these extra weights is provided by calculations of Doud and Pollack. We obtain theoretical evidence for n=4 by using automorphic inductions of Hecke characters
93 citations
TL;DR: In this article, a (G×(C×)l+1)-variety of complex symplectic groups is introduced, which is called the l-exotic nilpotent cone.
Abstract: Let G=Sp(2n,C) be a complex symplectic group. We introduce a (G×(C×)l+1)-variety Nl, which we call the l-exotic nilpotent cone. Then, we realize the Hecke algebra H of type Cn(1) with three parameters via equivariant algebraic K-theory in terms of the geometry of N2. This enables us to establish a Deligne-Langlands–type classification of simple H-modules under a mild assumption on parameters. As applications, we present a character formula and multiplicity formulas of H-modules
88 citations
TL;DR: In this paper, Gelfand and Kazhdan generalized the Luna slice theorem due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero.
Abstract: In the first part of this article, we generalize a descent technique due to Harish-Chandra to the case of a reductive group acting on a smooth affine variety both defined over an arbitrary local field F of characteristic zero. Our main tool is the Luna slice theorem. In the second part, we apply this technique to symmetric pairs. In particular, we prove that the pairs (GLn+k(F),GLn(F)×GLk(F)) and (GLn(E),GLn(F)) are Gelfand pairs for any local field F and its quadratic extension E. In the non-Archimedean case, the first result was proved earlier by Jacquet and Rallis [JR] and the second result was proved by Flicker [F]. We also prove that any conjugation-invariant distribution on GLn(F) is invariant with respect to transposition. For non-Archimedean F, the latter is a classical theorem of Gelfand and Kazhdan
TL;DR: In this paper, the authors give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V. They then prove that these sets can always be written as the intersection of V with a union of translates of tori of which we control the sum of the degrees.
Abstract: Let V be a subvariety of a torus defined over the algebraic numbers. We give a qualitative and quantitative description of the set of points of V of height bounded by invariants associated to any variety containing V . Especially, we determine whether such a set is or is not dense in V . We then prove that these sets can always be written as the intersection of V with a finite union of translates of tori of which we control the sum of the degrees. As a consequence, we prove a conjecture by the first author and David up to a logarithmic factor.
TL;DR: In this article, a long exact sequence for Legendrian submanifolds L⊂P×R was established, where P is an exact symplectic manifold, which admits a Hamiltonian isotopy that displaces the projection of L to P off of itself.
Abstract: We establish a long exact sequence for Legendrian submanifolds L⊂P×R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L to P off of itself. In this sequence, the singular homology H* maps to linearized contact cohomology CH*, which maps to linearized contact homology CH*, which maps to singular homology. In particular, the sequence implies a duality between Ker(CH*→H*) and CH*/Im(H*). Furthermore, this duality is compatible with Poincare duality in L in the following sense: the Poincare dual of a singular class which is the image of a∈CH* maps to a class α∈CH* such that α(a)=1. The exact sequence generalizes the duality for Legendrian knots in R3 (see [26]) and leads to a refinement of the Arnold conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [7]
TL;DR: In this paper, it was shown that the ideal of relations is generated by a set of quadric relations, with the single exception of the Segre cubic, which is generated in degree at most four and given an explicit description of the generators.
Abstract: A central question in invariant theory is that of determining the relations among invariants. Geometric invariant theory (GIT) quotients come with a natural ample line bundle and hence often a natural projective embedding. This question translates to determining the equations of the moduli space under this embedding. This article deals with one of the most classical quotients, the space of ordered points on the projective line. We show that under any weighting of the points, this quotient is cut out (scheme-theoretically) by a particularly simple set of quadric relations, with the single exception of the Segre cubic threefold, the space of six points with equal weight. We also show that the ideal of relations is generated in degree at most four, and we give an explicit description of the generators. If all the weights are even (e.g., in the case of equal weight for odd n), then we show that the ideal of relations is generated by quadrics
TL;DR: In this paper, the authors prove the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of PGLn(R).
Abstract: We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of PGLn(R). After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most O(� ǫ ) orbits of discriminant ≤ �. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We also give examples of sequences of periodic orbits of this action that fail to become equidistributed, even in higher rank. We give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees.
TL;DR: In this article, it was shown that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic and recover the group of G-equivariant automorphisms of X from these invariants.
Abstract: Let G be a connected reductive group. Recall that a homogeneous G-space X is called spherical if a Borel subgroup B ⊂ G has an open orbit on X. To X one assigns certain combinatorial invariants: the weight lattice, the valuation cone and the set of B-stable prime divisors. We prove that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic. Further, we recover the group of G-equivariant automorphisms of X from these invariants.
TL;DR: In this article, the authors studied the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space SL(n,R)/SO(n).
Abstract: In this article we study the asymptotic distribution of the cuspidal spectrum of arithmetic quotients of the symmetric space SL(n,R)/SO(n). In particular, we obtain Weyl's law with an estimation on the remainder term. This extends some of the main results of Duistermaat, Kolk, and Varadarajan ([DKV1]) to this setting
TL;DR: In this paper, it was shown that every Fourier-Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing.
Abstract: Every Fourier-Mukai equivalence between the derived categories of two K3 surfaces induces a Hodge isometry of their cohomologies viewed as Hodge structures of weight two endowed with the Mukai pairing. We prove that this Hodge isometry preserves the natural orientation of the four positive directions. This leads to a complete description of the action of the group of all autoequivalences on cohomology very much like the classical Torelli theorem for K3 surfaces determining all Hodge isometries that are induced by automorphisms
TL;DR: In this paper, the Stark-Heegnerner points are constructed over ring class fields of a modular, semistable elliptic curve of conductor N � (1) and satisfy a Shimura reciprocity law.
Abstract: Let F be a totally real field of narrow class number one, and let E/F be a modular, semistable elliptic curve of conductor N � (1) .L etK/F be a non-CM quadratic extension with (Disc K, N) = 1 such that the sign in the functional equation of L(E/K, s) is −1. Suppose further that there is a prime p|N that is inert in K .W e describe a p-adic construction of points on E which we conjecture to be rational over ring class fields of K/F and satisfy a Shimura reciprocity law. These points are expected to behave like classical Heegner points and can be viewed as new instances of the Stark-Heegner point construction of [5]. The key idea in our construction is a reinterpretation of Darmon’s theory of modular symbols and mixed period integrals in terms of group cohomology.
TL;DR: In this paper, the authors classify finite-dimensional simple spherical representations of rational double affine Hecke algebras, and study a remarkable family of finite dimensional simple spherical representation of double-affine hecke Algebraes.
Abstract: We classify finite-dimensional simple spherical representations of rational double affine Hecke algebras, and we study a remarkable family of finite-dimensional simple spherical representations of double affine Hecke algebras.
TL;DR: In this article, a general result on equality of weak limits of the zero counting measure, dνn, of orthogonal polynomials (defined by a measure dμ) and (1/n)Kn(x, x)dμ(x) was proved.
Abstract: We prove a general result on equality of the weak limits of the zero counting measure, dνn, of orthogonal polynomials (defined by a measure dμ) and (1/n)Kn(x, x)dμ(x). By combining this with the asymptotic upper bounds of Mate and Nevai [16] and Totik [33] on nλn(x), we prove some general results on ∫ Ι(1/n)Kn(x, x)dμs → 0 for the singular part of dμ and ∫ Ι |ρE(x) − (w(x)/n)Kn(x, x)| dx → 0, where ρE is the density of the equilibrium measure and w(x) the density of dμ.
TL;DR: In this article, the Castelnuovo-de Franchis inequality is extended to manifolds of arbitrary dimension, based on the theory of generic vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra.
Abstract: We extend to manifolds of arbitrary dimension the Castelnuovo-de Franchis inequality for surfaces. The proof is based on the theory of Generic Vanishing and higher regularity, and on the Evans-Griffith Syzygy Theorem in commutative algebra. Along the way we give a positive answer, in the setting of Kahler manifolds, to a question of Green-Lazarsfeld on the vanishing of higher direct images of Poincare' bundles. We indicate generalizations to arbitrary Fourier-Mukai transforms.
TL;DR: In this paper, the critical points of the length function on the free loop space Λ(M) of a compact Riemannian manifold M are the closed geodesics on M.
Abstract: The critical points of the length function on the free loop space Λ(M) of a compact Riemannian manifold M are the closed geodesics on M. The length function gives a filtration of the homology of Λ(M), and we show that the Chas-Sullivan product Hi(Λ)×Hj(Λ)→*Hi+j−n(Λ) is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring GrH*(Λ(M)) when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan's coproduct ∨ (see [Su1], [Su2])) on C*(Λ) as a product in cohomology Hi(Λ,Λ0)×Hj(Λ,Λ0)→⊛Hi+j+n−1(Λ,Λ0) (where Λ0=M is the constant loop). We show that ⊛ is also compatible with the length filtration, and we obtain a similar expression for the ring GrH*(Λ,Λ0). The nonvanishing of products σ*n and τ⊛n is shown to be determined by the rate at which the Morse index grows when a geodesic is iterated. We determine the full ring structure (H*(Λ,Λ0),⊛) for spheres M=Sn, n≥3
TL;DR: In this paper, the authors characterized the boundedness of planar directional maximal operators on a set of directions and showed that MΩ, the maximal operator associated to line segments in the directions Ω, is unbounded on Lp for all p 1.
Abstract: We completely characterize the boundedness of planar directional maximal operators on Lp. More precisely, if Ω is a set of directions, we show that MΩ, the maximal operator associated to line segments in the directions Ω, is unbounded on Lp for all p 1
TL;DR: In this article, the authors give a general method for establishing lower bounds for the representation dimension of given algebras or families of alges, and apply this method to most of the previous examples of large representation dimension for which the lower bound is improved to the correct value.
Abstract: The representation dimension is an invariant introduced by Auslander to measure how far a representation infinite algebra is from being representation finite. In 2005, Rouquier gave the first examples of algebras of representation dimension greater than three. Here we give the first general method for establishing lower bounds for the representation dimension of given algebras or families of algebras. The classes of algebras for which we explicitly apply this method include (but do not restrict to) most of the previous examples of algebras of large representation dimension, for some of which the lower bound is improved to the correct value
TL;DR: In this article, the authors derived direct images of the double tensor power and the general k-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases.
Abstract: Let X[n] be the Hilbert scheme of n points on the smooth quasi-projective surface X, and let L[n] be the tautological bundle on X[n] naturally associated to the line bundle L on X. As a corollary of Haiman's results, we express the image Φ(L[n]) of the tautological bundle L[n] for the Bridgeland-King-Reid equivalence Φ:Db(X[n])→DSnb(Xn) in terms of a complex CL• of Sn-equivariant sheaves in DSnb(Xn) and we characterize the image Φ(L[n]⊗⋅⋅⋅⊗L[n]) in terms of the hyperderived spectral sequence E1p,q associated to the derived k-fold tensor power of the complex CL•. The study of the Sn-invariants of this spectral sequence allows us to get the derived direct images of the double tensor power and of the general k-fold exterior power of the tautological bundle for the Hilbert-Chow morphism, providing Danila-Brion-type formulas in these two cases. This easily yields the computation of the cohomology of X[n] with values in L[n]⊗L[n] and ΛkL[n]
TL;DR: The main result of as discussed by the authors is that every closed Hamiltonian S 1 manifold is uniruled, i.e. it has a nonzero Gromov-Witten invariant one of whose constraints is a point.
Abstract: The main result of this note is that every closed Hamiltonian S 1 manifold is uniruled, i.e. it has a nonzero Gromov-Witten invariant one of whose constraints is a point. The proof uses the Seidel representation of 1 of the Hamiltonian group in the small quantum homology of M as well as the blow up technique recently introduced by Hu, Li and Ruan. It applies more generally to manifolds that have a loop of Hamiltonian symplectomorphisms with a nondegenerate fixed maximum. Some consequences for Hofer geometry are explored. An appendix discusses the structure of the quantum homology ring of uniruled manifolds.
TL;DR: In this article, it was shown that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2,R)-cocycle.
Abstract: We consider continuous SL(2,R)-cocycles over a strictly ergodic home- omorphism which fibers over an almost periodic dynamical system (generalized skew-shifts). We prove that any cocycle which is not uniformly hyperbolic can be approximated by one which is conjugate to an SO(2,R)-cocycle. Using this, we show that if a cocycle's homotopy class does not display a certain obstruc- tion to uniform hyperbolicity, then it can be C 0 -perturbed to become uniformly hyperbolic. For cocycles arising from Schrodinger operators, the obstruction van- ishes and we conclude that uniform hyperbolicity is dense, which implies that for a generic continuous potential, the spectrum of the corresponding Schrodinger operator is a Cantor set. 1. Statement of the Results
TL;DR: In this article, the authors introduce enumerative invariants of curves on Calabi-Yau 3-folds via certain stable objects in the derived category of coherent sheaves.
Abstract: In this article, we introduce new enumerative invariants of curves on Calabi-Yau 3-folds via certain stable objects in the derived category of coherent sheaves. We introduce the notion of limit stability on the category of perverse coherent sheaves, a subcategory in the derived category, and construct the moduli spaces of limit stable objects. We then define the counting invariants of limit stable objects using Behrend's constructible functions on those moduli spaces. It will turn out that our invariants are generalizations of counting invariants of stable pairs introduced by Pandharipande and Thomas. We will also investigate the wall-crossing phenomena of our invariants under change of stability conditions
TL;DR: In this paper, the Hitchin representation of open surface groups is generalized to arbitrary cross ratios. But the Hitchins are not generalized to PSL(n,R) and the cross ratios are not associated with the McShane-Mirzakhani identities.
Abstract: We generalise the McShane-Mirzakhani identities from hyperbolic geometry to arbitrary cross ratios. We define and study Hitchin representations of open surface groups to PSL(n,R). We associate cross ratios to these representations and then give explicit expressions for our generalised identities in terms of (a suitable choice of) Fock-Goncharov coordinates