559 research outputs found

    Euclidean Mahler measure and twisted links

    Full text link
    If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a parametrization of the 2-variable Homflypt polynomials of the corresponding links have bounded Mahler measure.Comment: This is the version published by Algebraic & Geometric Topology on 7 April 200

    Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. II: Higgs sheaves and admissible structures

    Full text link
    We study the basic properties of Higgs sheaves over compact K\"ahler manifolds and we establish some results concerning the notion of semistability; in particular, we show that any extension of semistable Higgs sheaves with equal slopes is semistable. Then, we use the flattening theorem to construct a regularization of any torsion-free Higgs sheaf and we show that it is in fact a Higgs bundle. Using this, we prove that any Hermitian metric on a regularization of a torsion-free Higgs sheaf induces an admissible structure on the Higgs sheaf. Finally, using admissible structures we proved some properties of semistable Higgs sheaves.Comment: 18 pages; some typos correcte

    Advances in the investigation of shock-induced reflectivity of porous carbon

    Get PDF
    AbstractWe studied the behavior of porous carbon compressed by laser-generated shock waves. In particular, we developed a new design for targets, optimized for the investigation of carbon reflectivity at hundred-GPa pressures and eV/k temperatures. Specially designed "two-layer-two materials" targets, comprising porous carbon on transparent substrates, allowed the probing of carbon reflectivity and a quite accurate determination of the position in the P, T plane. This was achieved by the simultaneous measurement of shock breakout times, sample temperature (by optical pyrometry) and uid velocity. The experiments proved the new scheme is reliable and appropriate for reflectivity measurements of thermodynamical states lying out of the standard graphite or diamond hugoniot. An increase of reflectivity in carbon has been observed at 260 GPa and 14,000 K while no increase in reflectivity is found at 200 GPa and 20,000 K. We also discuss the role of numerical simulations in the optimization of target parameters and in clarifying shock dynamics

    Proof of the Hyperplane Zeros Conjecture of Lagarias and Wang

    Full text link
    We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics and Fourier analysis.Comment: 25 page

    Analytic curves in algebraic varieties over number fields

    Full text link
    We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and P\'olya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and pp-adic curves). We also discuss geometric analogues, pertaining to the algebraic geometry of projective surfaces, of these arithmetic criteria.Comment: 55 pages. To appear in "Algebra, Arithmetic, and Geometry: In Honor of Y.i. Manin", Y. Tschinkel & Yu. Manin editors, Birkh\"auser, 200

    The Quantum McKay Correspondence for polyhedral singularities

    Get PDF
    Let G be a polyhedral group, namely a finite subgroup of SO(3). Nakamura's G-Hilbert scheme provides a preferred Calabi-Yau resolution Y of the polyhedral singularity C^3/G. The classical McKay correspondence describes the classical geometry of Y in terms of the representation theory of G. In this paper we describe the quantum geometry of Y in terms of R, an ADE root system associated to G. Namely, we give an explicit formula for the Gromov-Witten partition function of Y as a product over the positive roots of R. In terms of counts of BPS states (Gopakumar-Vafa invariants), our result can be stated as a correspondence: each positive root of R corresponds to one half of a genus zero BPS state. As an application, we use the crepant resolution conjecture to provide a full prediction for the orbifold Gromov-Witten invariants of [C^3/G].Comment: Introduction rewritten. Issue regarding non-uniqueness of conifold resolution clarified. Version to appear in Inventione

    Cohomological Hasse principle and motivic cohomology for arithmetic schemes

    Get PDF
    In 1985 Kazuya Kato formulated a fascinating framework of conjectures which generalizes the Hasse principle for the Brauer group of a global field to the so-called cohomological Hasse principle for an arithmetic scheme. In this paper we prove the prime-to-characteristic part of the cohomological Hasse principle. We also explain its implications on finiteness of motivic cohomology and special values of zeta functions.Comment: 47 pages, final versio
    corecore