2,786 research outputs found

    Low-momentum ring diagrams of neutron matter at and near the unitary limit

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    We study neutron matter at and near the unitary limit using a low-momentum ring diagram approach. By slightly tuning the meson-exchange CD-Bonn potential, neutron-neutron potentials with various 1S0^1S_0 scattering lengths such as as=−12070fma_s=-12070fm and +21fm+21fm are constructed. Such potentials are renormalized with rigorous procedures to give the corresponding asa_s-equivalent low-momentum potentials Vlow−kV_{low-k}, with which the low-momentum particle-particle hole-hole ring diagrams are summed up to all orders, giving the ground state energy E0E_0 of neutron matter for various scattering lengths. At the limit of as→±∞a_s\to \pm \infty, our calculated ratio of E0E_0 to that of the non-interacting case is found remarkably close to a constant of 0.44 over a wide range of Fermi-momenta. This result reveals an universality that is well consistent with the recent experimental and Monte-Carlo computational study on low-density cold Fermi gas at the unitary limit. The overall behavior of this ratio obtained with various scattering lengths is presented and discussed. Ring-diagram results obtained with Vlow−kV_{low-k} and those with GG-matrix interactions are compared.Comment: 9 pages, 7 figure

    Effective algebraic degeneracy

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    We prove that any nonconstant entire holomorphic curve from the complex line C into a projective algebraic hypersurface X = X^n in P^{n+1}(C) of arbitrary dimension n (at least 2) must be algebraically degenerate provided X is generic if its degree d = deg(X) satisfies the effective lower bound: d larger than or equal to n^{{(n+1)}^{n+5}}

    On the cohomology of pseudoeffective line bundles

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    The goal of this survey is to present various results concerning the cohomology of pseudoeffective line bundles on compact K{\"a}hler manifolds, and related properties of their multiplier ideal sheaves. In case the curvature is strictly positive, the prototype is the well known Nadel vanishing theorem, which is itself a generalized analytic version of the fundamental Kawamata-Viehweg vanishing theorem of algebraic geometry. We are interested here in the case where the curvature is merely semipositive in the sense of currents, and the base manifold is not necessarily projective. In this situation, one can still obtain interesting information on cohomology, e.g. a Hard Lefschetz theorem with pseudoeffective coefficients, in the form of a surjectivity statement for the Lefschetz map. More recently, Junyan Cao, in his PhD thesis defended in Grenoble, obtained a general K{\"a}hler vanishing theorem that depends on the concept of numerical dimension of a given pseudoeffective line bundle. The proof of these results depends in a crucial way on a general approximation result for closed (1,1)-currents, based on the use of Bergman kernels, and the related intersection theory of currents. Another important ingredient is the recent proof by Guan and Zhou of the strong openness conjecture. As an application, we discuss a structure theorem for compact K{\"a}hler threefolds without nontrivial subvarieties, following a joint work with F.Campana and M.Verbitsky. We hope that these notes will serve as a useful guide to the more detailed and more technical papers in the literature; in some cases, we provide here substantially simplified proofs and unifying viewpoints.Comment: 39 pages. This survey is a written account of a lecture given at the Abel Symposium, Trondheim, July 201

    Finite Generation of Canonical Ring by Analytic Method

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    In the 80th birthday conference for Professor LU Qikeng in June 2006 I gave a talk on the analytic approach to the finite generation of the canonical ring for a compact complex algebraic manifold of general type. This article is my contribution to the proceedings of that conference from my talk. In this article I give an overview of the analytic proof and focus on explaining how the analytic method handles the problem of infinite number of interminable blow-ups in the intuitive approach to prove the finite generation of the canonical ring. The proceedings of the LU Qikeng conference will appear as Issue No. 4 of Volume 51 of Science in China Series A: Mathematics (www.springer.com/math/applications/journal/11425)

    Section Extension from Hyperbolic Geometry of Punctured Disk and Holomorphic Family of Flat Bundles

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    The construction of sections of bundles with prescribed jet values plays a fundamental role in problems of algebraic and complex geometry. When the jet values are prescribed on a positive dimensional subvariety, it is handled by theorems of Ohsawa-Takegoshi type which give extension of line bundle valued square-integrable top-degree holomorphic forms from the fiber at the origin of a family of complex manifolds over the open unit 1-disk when the curvature of the metric of line bundle is semipositive. We prove here an extension result when the curvature of the line bundle is only semipositive on each fiber with negativity on the total space assumed bounded from below and the connection of the metric locally bounded, if a square-integrable extension is known to be possible over a double point at the origin. It is a Hensel-lemma-type result analogous to Artin's application of the generalized implicit function theorem to the theory of obstruction in deformation theory. The motivation is the need in the abundance conjecture to construct pluricanonical sections from flatly twisted pluricanonical sections. We also give here a new approach to the original theorem of Ohsawa-Takegoshi by using the hyperbolic geometry of the punctured open unit 1-disk to reduce the original theorem of Ohsawa-Takegoshi to a simple application of the standard method of constructing holomorphic functions by solving the d-bar equation with cut-off functions and additional blowup weight functions

    Unitarity potentials and neutron matter at the unitary limit

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    We study the equation of state of neutron matter using a family of unitarity potentials all of which are constructed to have infinite 1S0^1S_0 scattering lengths asa_s. For such system, a quantity of much interest is the ratio ξ=E0/E0free\xi=E_0/E_0^{free} where E0E_0 is the true ground-state energy of the system, and E0freeE_0^{free} is that for the non-interacting system. In the limit of as→±∞a_s\to \pm \infty, often referred to as the unitary limit, this ratio is expected to approach a universal constant, namely ξ∼0.44(1)\xi\sim 0.44(1). In the present work we calculate this ratio ξ\xi using a family of hard-core square-well potentials whose asa_s can be exactly obtained, thus enabling us to have many potentials of different ranges and strengths, all with infinite asa_s. We have also calculated ξ\xi using a unitarity CDBonn potential obtained by slightly scaling its meson parameters. The ratios ξ\xi given by these different unitarity potentials are all close to each other and also remarkably close to 0.44, suggesting that the above ratio ξ\xi is indifferent to the details of the underlying interactions as long as they have infinite scattering length. A sum-rule and scaling constraint for the renormalized low-momentum interaction in neutron matter at the unitary limit is discussed.Comment: 7.5 pages, 7 figure

    On Modeling Credit Defaults: A Probabilistic Boolean Network Approach

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    Local syzygies of multiplier ideals

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    In recent years, multiplier ideals have found many applications in local and global algebraic geometry. Because of their importance, there has been some interest in the question of which ideals on a smooth complex variety can be realized as multiplier ideals. Other than integral closure no local obstructions have been known up to now, and in dimension two it was established by Favre-Jonsson and Lipman-Watanabe that any integrally closed ideal is locally a multiplier ideal. We prove the somewhat unexpected result that multiplier ideals in fact satisfy some rather strong algebraic properties involving higher syzygies. It follows that in dimensions three and higher, multiplier ideals are very special among all integrally closed ideals.Comment: 8 page

    Positivity of relative canonical bundles and applications

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    Given a family f:X→Sf:\mathcal X \to S of canonically polarized manifolds, the unique K\"ahler-Einstein metrics on the fibers induce a hermitian metric on the relative canonical bundle KX/S\mathcal K_{\mathcal X/S}. We use a global elliptic equation to show that this metric is strictly positive on X\mathcal X, unless the family is infinitesimally trivial. For degenerating families we show that the curvature form on the total space can be extended as a (semi-)positive closed current. By fiber integration it follows that the generalized Weil-Petersson form on the base possesses an extension as a positive current. We prove an extension theorem for hermitian line bundles, whose curvature forms have this property. This theorem can be applied to a determinant line bundle associated to the relative canonical bundle on the total space. As an application the quasi-projectivity of the moduli space Mcan\mathcal M_{\text{can}} of canonically polarized varieties follows. The direct images Rn−pf∗ΩX/Sp(KX/S⊗m)R^{n-p}f_*\Omega^p_{\mathcal X/S}(\mathcal K_{\mathcal X/S}^{\otimes m}), m>0m > 0, carry natural hermitian metrics. We prove an explicit formula for the curvature tensor of these direct images. We apply it to the morphisms SpTS→Rpf∗ΛpTX/SS^p \mathcal T_S \to R^pf_*\Lambda^p\mathcal T_{\mathcal X/S} that are induced by the Kodaira-Spencer map and obtain a differential geometric proof for hyperbolicity properties of Mcan\mathcal M_{\text{can}}.Comment: Supercedes arXiv:0808.3259v4 and arXiv:1002.4858v2. To appear in Invent. mat
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