Showing papers in "American Journal of Mathematics in 2006"
TL;DR: In this article, the intersection operator was shown to be the so-called intersection operator, an operator that played a critical role in the solution of the Busemann-Petty problem.
Abstract: All GL (n) covariant star-body-valued valuations on convex polytopes are completely classified. It is shown that there is a unique nontrivial such valuation. This valuation turns out to be the so-called "intersection operator"—an operator that played a critical role in the solution of the Busemann-Petty problem.
187 citations
TL;DR: In this article, the authors studied the algebraic degree of the critical equations of the problem of maximizing a product of powers of polynomials, and showed that the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms.
Abstract: Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses, the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms. Exact formulae in terms of degrees and Newton polytopes are given for polynomials with generic coefficients.
141 citations
TL;DR: The affine Schur functions as discussed by the authors generalize the k-Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur function of Postnikov.
Abstract: We define a new family [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/] of generating functions for w ∈ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i"/] which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity properties in terms of a subfamily of symmetric functions called affine Schur functions. As applications, we show how affine Stanley symmetric functions generalize the (dual of the) k -Schur functions of Lapointe, Lascoux and Morse as well as the cylindric Schur functions of Postnikov. Conjecturally, affine Stanley symmetric functions should be related to the cohomology of the affine flag variety.
124 citations
TL;DR: In this paper, it was shown that a cuspidal automorphic representation on GLn is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor.
Abstract: We establish zero-free regions tapering as an inverse power of the analytic conductor for Rankin-Selberg L-functions on GLn × GLn . Such zero-free regions are equivalent to commensurate lower bounds on the edge of the critical strip, and in the case of L(s, π × π), on the residue at s =1 . As an application we show that a cuspidal automorphic representation on GLn is determined by a finite number of its Dirichlet series coefficients, and that this number grows at most polynomially in the analytic conductor. Introduction. Let A be the ring of adeles over a number field F and let π and π � be two cuspidal representations of GLn(A) with restricted tensor product decompositions π = ⊗vπv and π � = ⊗vπ �
107 citations
TL;DR: The Strichartz inequalities for Riemannian manifolds were obtained by Tataru et al. as mentioned in this paper for the case of asymptotically conic manifold M with either short-range or long-range metric perturbation.
Abstract: We obtain the Strichartz inequalities ║u║ L q t L r x ([0,1]× M ) ≥ C║ u (0) L 2 ║( M ) for any smooth n -dimensional Riemannian manifold M which is asymptotically conic at infinity (with either short-range or long-range metric perturbation) and nontrapping, where u is a solution to the Schrodinger equation iu t + 1/2 Δ M u = 0, and 2 q, r ≥ ∞ are admissible Strichartz exponents (2/ q + n/r = n /2). This corresponds with the estimates available for Euclidean space (except for the endpoint ( q, r ) = (2, 2 n/n -2) when n 2). These estimates imply existence theorems for semi-linear Schrodinger equations on M , by adapting arguments from Cazenave and Weissler and Kato. This result improves on our previous result, which was an L 4 t,x Strichartz estimate in three dimensions. It is closely related to results of Staffilani-Tataru, Burq, Robbiano-Zuily and Tataru, who consider the case of asymptotically flat manifolds.
103 citations
TL;DR: In this article, a number of necessary and sufficient conditions for pure discrete spectrum were developed, including injectivity of the canonical torus map (the geometric realization), Geometric Coincidence Condition, (partial) commutation of T and the dual R d -1 -action, measure and tiling properties of Rauzy fractals, and concrete algorithms.
Abstract: We are concerned with the tiling flow T associated to a substitution φ over a finite alphabet. Our focus is on substitutions that are unimodular Pisot, i.e., their matrix is unimodular and has all eigenvalues strictly inside the unit circle with the exception of the Perron eigenvalue λ 1. The motivation is provided by the (still open) conjecture asserting that T has pure discrete spectrum for any such φ. We develop a number of necessary and sufficient conditions for pure discrete spectrum, including: injectivity of the canonical torus map (the geometric realization), Geometric Coincidence Condition, (partial) commutation of T and the dual R d -1 -action, measure and tiling properties of Rauzy fractals, and concrete algorithms. Some of these are original and some have already appeared in the literature-as sufficient conditions only-but they all emerge from a unified approach based on the new device: the strand space F φ of φ. The proof of the necessity hinges on determination of the discrete spectrum of T as that of the associated Kronecker toral flow.
100 citations
TL;DR: The Weil-Petersson metric on Teichmiiller space is Gromov-hyperbolic if and only if d(S) 3 as discussed by the authors, where S is a surface with genus g and n boundary components.
Abstract: Let 5 be a surface with genus g and n boundary components, and let d(S) = 3g - 3 + n denote the number of curves in any pants decomposition of S. We employ metric properties of the graph of pants decompositions P(S) to prove that the Weil-Petersson metric on Teichmiiller space Teich (S) is Gromov-hyperbolic if and only if d(S) 3, the Weil-Petersson metric has higher rank in the sense of Gromov (it admits a quasi-isometric embedding of Rk,k > 2); when d(S) 3, and that no complete Riemannian metric of pinched negative curvature exists on Moduli space M(S) when d(S) > 2. 1. Introduction. The Weil-Petersson metric on Teichmiiller space Teich (5) has many curious properties. It is a Riemannian metric with negative sectional curvature, but its curvatures are not bounded away from zero or negative infinity. It is geodesically convex, but it is not complete. In this paper we show that in spite of exhibiting negative curvature behavior, the Weil-Petersson metric is not coarsely negatively curved except for topologically simple surfaces S. Our main theorem answers a question of Bowditch (Be, Question 11.4).
93 citations
TL;DR: In this paper, the authors give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 ≤ 1, and apply the results to syzygies, Grobner bases, products and powers of ideals.
Abstract: Let S = K(x1, ... , xn), let A, B be finitely generated graded S-modules, and let m = (x1, ... , xn) ⊂ S. We give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 (A, B) ≤ 1. We apply the results to syzygies, Grobner bases, products and powers of ideals, and to the relationship of the Rees and symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first � (n − 1)/2� steps of the resolution of I are linear, then I 2 is a power of m.
93 citations
TL;DR: Theorem 7.1 as discussed by the authors establishes the nontrivial bound Σ n ≤ T a n e 2 π i n α = O e (T 3/4+e ), uniformly in α ∈ R, for a n the coefficients of the L -function of a cusp form on GL (3, Z)\ GL(3, R), and derives an equivalence between analogous cancellation statements for cusp forms on GL( n, R) and the sizes of certain period integrals.
Abstract: In a previous paper with Schmid we considered the regularity of automorphic distributions for GL (2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound Σ n ≤ T a n e 2 π i n α = O e ( T 3/4+e ), uniformly in α ∈ R, for a n the coefficients of the L -function of a cusp form on GL (3, Z)\ GL (3,R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL ( n ,R), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form L -functions.
88 citations
TL;DR: In this paper, a generalization of Demailly-Ein-Lazarsfeld's subadditivity formula and Mustaţǎ's summation formula for multiplier ideals to the case of singular varieties, using characteristic p methods, is presented.
Abstract: We prove a generalization of Demailly-Ein-Lazarsfeld's subadditivity formula and Mustaţǎ's summation formula for multiplier ideals to the case of singular varieties, using characteristic p methods. As an application of our formula, we improve Hochster-Huneke's result on the growth of symbolic powers of ideals in singular affine algebras.
80 citations
TL;DR: In this paper, the authors give a precise characterization of those plurisubharmonic functions for which one can well define the Monge-Ampere operator as a regular Borel measure.
Abstract: We give a precise characterization of those plurisubharmonic functions for which one can well define the Monge-Ampere operator as a regular Borel measure.
TL;DR: In this paper, the notions of Betti numbers and Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism φ: R → S.
Abstract: The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S , for some local homomorphism φ: R → S . Various techniques are developed to study the new invariants and to establish their basic properties. In some cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite R -modules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R . A third is to obtain criteria for a ring equipped with a "contracting" endomorphism—such as the Frobenius endomorphism—to be regular or complete intersection; these results represent broad generalizations of Kunz's characterization of regularity in prime characteristic.
TL;DR: It is shown that the L.p. norm bounds proven by Sogge in the case of smooth metrics hold under this limited regularity assumption.
Abstract: In this paper, we establish L p norm bounds for spectral clusters on compact manifolds, under the assumption that the metric is C 1,1 . Precisely, we show that the L p estimates proven by Sogge in the case of smooth metrics hold under this limited regularity assumption. It is known by examples of Smith-Sogge that such estimates fail for C 1,α metrics if α < 1.
TL;DR: In this paper, it was shown that the characteristic function of the unit disc in R 2 is the Fourier multiplier of a bounded bilinear operator from L p 1 (R ) × L p 2 (R) into L p (R ), when 2 ≤ p 1, p 2 < ∞ and [inline-graphic xmlns:xlink="http://www.w3.org/xlink" xlink:href="01i"/
Abstract: A classical theorem of C. Fefferman says that the characteristic function of the unit disc is not a Fourier multiplier on L p ( R 2 ) unless p = 2. In this article we obtain a result that brings a contrast with the previous theorem. We show that the characteristic function of the unit disc in R 2 is the Fourier multiplier of a bounded bilinear operator from L p 1 ( R ) × L p 2 ( R ) into L p ( R ), when 2 ≤ p 1 , p 2 < ∞ and [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i"/]. The proof of this result is based on a new decomposition of the unit disc and delicate orthogonality and combinatorial arguments. This result implies norm convergence of bilinear Fourier series and strengthens the uniform boundedness of the bilinear Hilbert transforms, as it yields uniform vector-valued bounds for families of bilinear Hilbert transforms.
TL;DR: In this article, the authors study Hessenberg subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions and provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space.
Abstract: We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GL n (C) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules generalizing the Eulerian numbers. We give an equivalent formulation of this result in terms of roots. We conclude with a section on open questions.
TL;DR: Asymptotic cohomological functions as discussed by the authors generalize the concept of the volume of a line bundle to higher cohomology and give a notion invariant under the numerical equivalence of divisors, and extend uniquely to continuous functions on the real Neron-Severi space.
Abstract: We consider certain cohomological invariants called asymptotic cohomological functions, which are associated to irreducible projective varieties. Asymptotic cohomological functions are generalizations of the concept of the volume of a line bundle-the asymptotic growth of the number of global sections-to higher cohomology. We establish that they give a notion invariant under the numerical equivalence of divisors, and extend uniquely to continuous functions on the real Neron-Severi space. To illustrate the theory, we work out these invariants for abelian varieties, smooth surfaces, and certain homogeneous spaces.
TL;DR: In this paper, the authors proved a long time existence result for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, which generalizes a preceding result concerning the case of spheres, obtained in an earlier paper by the authors.
Abstract: We prove a long time existence result for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds. This generalizes a preceding result concerning the case of spheres, obtained in an earlier paper by the authors. The proof relies on almost orthogonality properties of products of eigenfunctions of positive elliptic selfadjoint operators on a compact manifold and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.
TL;DR: In this article, the authors showed that the three-dimensional Schrodinger propogator e itH is a bounded map from L 1 to L ∞ with norm controlled by | t | -3/2 provided the potential satisfies two conditions: an integrability condition limiting the singularities and decay of V, and a zero-energy spectral condition on H.
Abstract: The three-dimensional Schrodinger propogator e itH , H = -Δ + V , is a bounded map from L 1 to L ∞ with norm controlled by | t | -3/2 provided the potential satisfies two conditions: An integrability condition limiting the singularities and decay of V , and a zero-energy spectral condition on H . This is shown by expressing the spectral measure of H in terms of its resolvents and proving a family of L p mapping estimates for the resolvents. Previous results in this direction had required V to satisfy explicit pointwise bounds.
TL;DR: In this paper, the authors consider minimal compact complex surfaces S with Betti numbers b\ = 1 and n = bi > 0 and prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces.
Abstract: We consider minimal compact complex surfaces S with Betti numbers b\ = 1 and n = bi > 0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m > 1 and a flat line bundle F such that H®(S,-mK 0; these surfaces admit no nonconstant mero-morphic functions. The major problem in classification of non-kahlerian surfaces is to achieve the classification of surfaces S of class VIIJ. All known surfaces of this class contain Global Spherical Shells (GSS), i.e., admit a biholomorphic map ip\ U-> V from a neighbourhood U C C2\ {0} of the sphere 53 = dB2 onto an open set V such that I = (f(S3) does not disconnect S. Are there other surfaces ? In first section we investigate the general situation: A theorem of Donaldson [13] gives a Z-base (£,) of //2(S,Z), such that £/£, =- B of S these line bundles form families £/. We propose the following conjecture which can be easily checked for surfaces with GSS:
TL;DR: In this article, a coarse lower bound for L-functions of Langlands-Shahidi type of generic cuspidal automorphic representations on the line Re (s) = 1 was proved.
Abstract: We prove a coarse lower bound for L-functions of Langlands-Shahidi type of generic cuspidal automorphic representations on the line Re (s) = 1. We follow the path suggested by Sarnak using Eisenstein series and the Maass-Selberg relations. The bounds are weaker than what the method of de la Vallee Poussin gives for the standard L-functions of GLn, but are applicable to more general automorphic L-functions. Our Theorem answers in a strong form a conjecture posed by Gelbart and Shahidi (J. Amer. Math. Soc. 14 (2001)), and sharpens and considerably simplifies the proof of the main result of that paper.
TL;DR: In this paper, the authors obtained results on approximation of holomorphic maps by algebraic maps, the jet transversality theorem for holomorphic and algebraic mapping between certain classes of manifolds, and the homotopy principle for holomorph submersions of Stein manifolds to certain algebraic manifolds.
Abstract: We obtain results on approximation of holomorphic maps by algebraic maps, the jet transversality theorem for holomorphic and algebraic maps between certain classes of manifolds, and the homotopy principle for holomorphic submersions of Stein manifolds to certain algebraic manifolds.
TL;DR: In this paper, the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and Fourier coefficients of Siegel modular forms was generalized to the case where the cycles have local coefficients.
Abstract: The purpose of this paper is to generalize the relation between intersection numbers of cycles in locally symmetric spaces of orthogonal type and Fourier coefficients of Siegel modular forms to the case where the cycles have local coefficients. Now the correspondence will involve vector-valued Siegel modular forms.
TL;DR: In this article, it was shown that if X ⊂ P r is any 2-regular scheme (in the sense of Castelnuovo-Mumford) then X is small.
Abstract: We prove that if X ⊂ P r is any 2-regular scheme (in the sense of Castelnuovo-Mumford) then X is small. This means that if L is a linear space and Y := L ∩ X is finite, then Y is linearly independent in the sense that the dimension of the linear span of Y is deg Y + 1. The converse is true and well-known for finite schemes, but false in general. The main result of this paper is that the converse, "small implies 2-regular", is also true for reduced schemes (algebraic sets). This is proven by means of a delicate geometric analysis, leading to a complete classification: we show that the components of a small algebraic set are varieties of minimal degree, meeting in a particularly simple way. From the classification one can show that if X ⊂ P r is 2-regular, then so is Xred, and so also is the projection of X from any point of X. Our results extend the Del Pezzo-Bertini classification of varieties of minimal degree, the charac- terization of these as the varieties of regularity 2 by Eisenbud-Goto, and the construction of 2-regular square-free monomial ideals by Froberg. 0. Introduction Throughout this paper we will work with projective schemes X ⊂ P r over an algebraically closed field k. The (Castelnuovo-Mumford) regularity of X ⊂ P r is a basic homological measure of the complexity of X and its embedding in P r that gives a bound for the degrees of the generators of the defining ideal IX of X and for many other invariants. The only schemes of regular- ity 1 are the linear spaces; but no classification is known for projective schemes of regularity 2. In this paper we prove a structure theorem for reduced 2-regular schemes, showing that their irreducible components are varieties of minimal degree and characterizing how these components can meet. We also show that the reduced structure on any 2-regular scheme is 2-regular, and thus we obtain a complete description of the reduced structures on 2-regular schemes. (Since a high Veronese re-embedding of any zero-dimensional scheme is 2-regular, one cannot hope to characterize the isomorphism types of all 2-regular nonreduced schemes.) Before stating our results we review some basic notions. For any subscheme X ⊂ P r we write span(X) for the smallest linear subspace of P r containing X. Recall that every variety (≡ reduced irreducible scheme) X ⊂ P r satisfies the condition deg (X) ≥ 1 + codim (X, span(X)) (*)
TL;DR: In this paper, the authors studied the center focus problem for polynomial vector fields whose integral trajectories are closed curves with interiors containing a fixed point, a center, which is closely connected to the classical Poincare Center-Focus problem about the characterization of planar vector fields.
Abstract: We study the Center Problem for equations v ' = Σ ∞ i =1 a i (x)v i +1 . This problem is closely connected to the classical Poincare Center-Focus problem about the characterization of planar polynomial vector fields whose integral trajectories are closed curves with interiors containing a fixed point, a center .
TL;DR: In this article, it was shown that the T-equivariant cohomology of X = G/P satisfies a positivity property: its structure constants are nonnegative integers equal to the number of intersection points of three Schubert varieties, in general position, whose codimensions add up to the dimension of X.
Abstract: A conjecture of D. Peterson, proved by W. Graham, states that the structure constants of the (T-)equivariant cohomology of a homogeneous space G/P satisfy a certain positivity property. In this paper we show that this positivity property holds in the more general situation of equivariant quantum cohomology. 1. Introduction. It is well known that the (integral) cohomology of the homogeneous space X = G/P (for G a connected, semisimple, complex Lie group and P a parabolic subgroup) satisfies a positivity property: its structure constants are nonnegative integers equal to the number of intersection points of three Schubert varieties, in general position, whose codimensions add up to the dimension of X. Recently, Graham (Gr) has proved a conjecture of Peterson (P), asserting that H � (X), the T-equivariant cohomology of X, where T � (C � ) r is a maximal torus in G, enjoys a more general positivity property.
Abstract: We prove C 1,γ regularity of Lipschitz free boundaries of two-phase problems for a class of homogeneous fully nonlinear elliptic operators F (D 2 u ( x ), x ) with Holder dependence on x , containing convex (concave) operators.
TL;DR: In this article, the authors give new formulas for Grothendieck polynomials of two types, one expressing any specialization of a Grothmann polynomial in at least two sets of variables as a linear combination of products of Grothman polynoms in each set of variables, with coefficients Schubert structure constants.
Abstract: We give new formulas for Grothendieck polynomials of two types. One type expresses any specialization of a Grothendieck polynomial in at least two sets of variables as a linear combination of products of Grothendieck polynomials in each set of variables, with coefficients Schubert structure constants for Grothendieck polynomials. The other type is in terms of chains in the Bruhat order. We compare this second type to other constructions of Grothendieck polynomials within the more general context of double Grothendieck polynomials and the closely related H-polynomials. Our methods are based upon the geometry of permutation patterns.
TL;DR: In this article, the authors extend the notion of symplectic implosion to the category of quasi-Hamiltonian K -manifolds, where K is a simply connected compact Lie group and the imploded cross-section of the double K × K turns out to be universal in a suitable sense.
Abstract: The purpose of this paper is twofold. First we extend the notion of symplectic implosion to the category of quasi-Hamiltonian K -manifolds, where K is a simply connected compact Lie group. The imploded cross-section of the double K × K turns out to be universal in a suitable sense. It is a singular space, but some of its strata have a nonsingular closure. This observation leads to interesting new examples of quasi-Hamiltonian K -manifolds, such as the "spinning 2 n -sphere" for K = SU ( n ). Secondly we construct a universal ("master") moduli space of parabolic bundles with structure group K over a marked Riemann surface. The master moduli space carries a natural action of a maximal torus of K and a torus-invariant stratification into manifolds, each of which has a symplectic structure. An essential ingredient in the construction is the universal implosion. Paradoxically, although the universal implosion has no complex structure (it is the four-sphere for K = SU (2)), the master moduli space turns out to be a complex algebraic variety.
TL;DR: In this paper, the authors generalize Donaldson, Gieseker, Li and O'Grady's results on generic smoothness of the moduli spaces of sheaves with fixed determinant and large discriminant to positive characteristic.
Abstract: We generalize Donaldson, Gieseker, Li and O'Grady's results on generic smoothness of the moduli spaces of sheaves with fixed determinant and large discriminant to positive characteristic. We also show optimal bounds on the Castelnuovo-Mumford regularity of sheaves on surfaces and we use it to give the first general effective results on irreducibility of the moduli spaces.
TL;DR: In this paper, it was shown that a finite mapping is transversal to the target manifold provided this manifold is of finite type, which is the case of generic manifolds of higher codimension.
Abstract: We prove here new results about transversality and related geometric properties of a holomorphic, formal, or CR mapping, sending one generic submanifold of C^ into another. One of our main results is that a finite mapping is transversal to the target manifold provided this manifold is of finite type. For the case of hypersurfaces, transversality in this context was proved by Baouendi and the second author in 1990. The general case of generic manifolds of higher codimension, which we treat in this paper, had remained an open problem since then. Applications of this result include a sufficient condition for a finite mapping to be a local diffeomorphism.