Showing papers in "International Journal of Mathematics in 2013"
TL;DR: In this article, the existence of real hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3.
Abstract: We classify real hypersurfaces with isometric Reeb flow in the complex quadrics Qm = SOm+2/SOmSO2, m ≥ 3. We show that m is even, say m = 2k, and any such hypersurface is an open part of a tube around a k-dimensional complex projective space ℂPk which is embedded canonically in Q2k as a totally geodesic complex submanifold. As a consequence, we get the non-existence of real hypersurfaces with isometric Reeb flow in odd-dimensional complex quadrics Q2k+1, k ≥ 1. To our knowledge the odd-dimensional complex quadrics are the first examples of homogeneous Kahler manifolds which do not admit a real hypersurface with isometric Reeb flow.
64 citations
TL;DR: In this article, twisted cycles with respect to an Euler-type integral representation of Lauricella's hypergeometric function were constructed, which correspond to 2m linearly independent solutions to the system of differential equations annihilating FC.
Abstract: We study Lauricella's hypergeometric function FC by using twisted (co)homology groups. We construct twisted cycles with respect to an Euler-type integral representation of FC. These cycles correspond to 2m linearly independent solutions to the system of differential equations annihilating FC. Using intersection forms of twisted (co)homology groups, we obtain twisted period relations which give quadratic relations for Lauricella's FC.
37 citations
TL;DR: In this paper, it was shown that the Fubini-study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvatures is strictly positive.
Abstract: We discuss positive closed currents and Fubini–Study currents on orbifolds, as well as Bergman kernels of singular Hermitian orbifold line bundles. We prove that the Fubini–Study currents associated to high powers of a semipositive singular line bundle converge weakly to the curvature current on the set where the curvature is strictly positive, generalizing a well-known theorem of Tian. We include applications to the asymptotic distribution of zeros of random holomorphic sections.
26 citations
TL;DR: In this article, it was shown that any metric attached to a klt pair (X, D) has cone singularities along D on the log-smooth locus of the pair, under some technical assumption on the cone angles.
Abstract: We prove that any Kahler–Einstein metric attached to a klt pair (X, D) has cone singularities along D on the log-smooth locus of the pair, under some technical assumption on the cone angles.
26 citations
TL;DR: In this article, the authors present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups and apply this machinery to find a concrete realization of the Poisson boundary of the compact quantum group SUq(2) arising from measures on its spectrum.
Abstract: We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SUq(2) arising from measures on its spectrum.
25 citations
TL;DR: In this article, a pseudo-trace function for vertex operator algebras satisfying Zhu's finiteness condition was proposed and applied to the ℤ2-orbifold model associated with d-pairs of symplectic fermions.
Abstract: We propose a method to give pseudo-trace functions for vertex operator algebras satisfying Zhu's finiteness condition and apply our method to the ℤ2-orbifold model associated with d-pairs of symplectic fermions. Pseudo-trace functions are defined by using symmetric linear functions on (higher) Zhu's algebras. But our approach does not use such algebras. For d = 1, we determine the dimension of the space of one-point functions, that is, the one of the triplet -algebra. For d > 1, we construct 22d-1 + 3 linearly independent one-point functions and study their values at the vacuum vector.
23 citations
TL;DR: In this article, the monodromy action on the mod 2 cohomology for SL(2, ℂ) Hitchin systems is analyzed in terms of the moduli space of semistable SL( 2, ε)-Higgs bundles.
Abstract: We calculate the monodromy action on the mod 2 cohomology for SL(2, ℂ) Hitchin systems and give an application of our results in terms of the moduli space of semistable SL(2, ℝ)-Higgs bundles.
22 citations
TL;DR: In this article, an area-preserving flow for convex plane curves is presented, which decreases the perimeter of the evolving curve and makes the curve more and more circular during the evolution process and finally, as t goes to infinity, the limiting curve will be a finite circle in the C∞ metric.
Abstract: Motivated by Gage [On an area-preserving evolution equation for plane curves, in Nonlinear Problems in Geometry, ed D M DeTurck, Contemporary Mathematics, Vol 51 (American Mathematical Society, Providence, RI, 1986), pp 51–62] and Ma–Cheng [A non-local area preserving curve flow, preprint (2009), arXiv:09071430v2, [mathDG]], in this paper, an area-preserving flow for convex plane curves is presented This flow will decrease the perimeter of the evolving curve and make the curve more and more circular during the evolution process And finally, as t goes to infinity, the limiting curve will be a finite circle in the C∞ metric
20 citations
TL;DR: In this paper, a class of inner automorphisms of compact quantum groups (CQGs) is introduced and the behavior of normal subgroups of CQGs under these automomorphisms is explored.
Abstract: We introduce a class of automorphisms of compact quantum groups (CQGs) which may be thought of as inner automorphisms and explore the behavior of normal subgroups of CQGs under these automorphisms. We also define the notion of center of a CQG and compute the center for several examples. We briefly touch upon the commutator subgroup of a CQG and discuss how its relation with the center can be different from the classical case.
19 citations
TL;DR: In this paper, the universal character ring of the (-2, 2m + 1, 2n)-pretzel link was shown to be reduced for all integers m and n.
Abstract: We explicitly calculate the universal character ring of the (-2, 2m + 1, 2n)-pretzel link and show that it is reduced for all integers m and n.
19 citations
TL;DR: In this article, the Fubini theorems for integral transforms and convolution products for functionals on function space were established, and they were used to establish the FUBINI theorem for convolutional functions.
Abstract: In this paper we establish Fubini theorems for integral transforms and convolution products for functionals on function space.
TL;DR: In this article, the Stein-Weiss inequality for the B-Riesz potential generated by the Riemann-Liouville operator was established, and the Pitt's and Beckner logarithmic inequalities related to the connected Fourier transform were proved.
Abstract: First, we establish the Stein–Weiss inequality for the B-Riesz potential generated by the Riemann–Liouville operator. Next, we prove the Pitt's and Beckner logarithmic inequalities related to the connected Fourier transform.
TL;DR: In this paper, a construction of σ-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface is proposed.
Abstract: Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular forms have pointed to the relevance of τ-functions, which are in turn connected with a specific type of abelian function, the (Kleinian) σ-function. Klein originally generalized Weierstrass' σ-function to hyperelliptic curves in a geometric way, then from a modular point of view to trigonal curves. Recently a modular generalization for all curves was given, as well as the geometric one for certain affine planar curves, known as (n, s) curves, and their generalizations known as "telescopic" curves. This paper proposes a construction of σ-functions based on the nature of the Weierstrass semigroup at one point of the Riemann surface as a generalization of the construction for plane affine models of the Riemann surface. Our examples are not telescopic. Because our definition is algebraic, in the sense of being related to the field of meromorphic functions on the curve, we are able to consider properties of the σ-functions such as Jacobi inversion formulae, and to observe relationships between their properties and those of a Norton basis for replicable functions, in turn relevant to the Monstrous Moonshine.
TL;DR: In this article, the Nielsen-Kempe separability criterion for quantum information theory is shown to hold for positive definite matrices written in β × β Hermitian blocks and β = 2p for some p ∈ ℕ.
Abstract: Let H = [As, t] be a positive definite matrix written in β × β Hermitian blocks and let Δ = A1, 1 + ⋯ + Aβ, β be its partial trace. Assume that β = 2p for some p ∈ ℕ. Then, up to a direct sum operation, H is the average of β matrices isometrically congruent to Δ. A few corollaries are given, related to important inequalities in quantum information theory such as the Nielsen–Kempe separability criterion.
TL;DR: In this paper, a general framework for modified expected dimensions for rank-2 Brill-Noether loci with prescribed special determinants is introduced, which applies a priori for arbitrary rank, and use it to prove modified expected dimension bounds in several new cases, applying both to rank 2 and to higher rank.
Abstract: Extending our previous study of modified expected dimensions for rank-2 Brill–Noether loci with prescribed special determinants, we introduce a general framework which applies a priori for arbitrary rank, and use it to prove modified expected dimension bounds in several new cases, applying both to rank 2 and to higher rank. The main tool is the introduction of generalized alternating Grassmannians, which are the loci inside Grassmannians corresponding to subspaces which are simultaneously isotropic for a family of multilinear alternating forms on the ambient vector space. In the case of rank 2 with two-dimensional spaces of sections, we adapt arguments due to Teixidor i Bigas to show that our new modified expected dimensions are in fact sharp.
TL;DR: In this paper, the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad(K)-submodules was constructed.
Abstract: We construct the homogeneous Einstein equation for generalized flag manifolds G/K of a compact simple Lie group G whose isotropy representation decomposes into five inequivalent irreducible Ad(K)-submodules. To this end, we apply a new technique which is based on a fibration of a flag manifold over another such space and the theory of Riemannian submersions. We classify all generalized flag manifolds with five isotropy summands, and we use Grobner bases to study the corresponding polynomial systems for the Einstein equation. For the generalized flag manifolds E6/(SU(4) × SU(2) × U(1) × U(1)) and E7/(U(1) × U(6)) we find explicitly all invariant Einstein metrics up to isometry. For the generalized flag manifolds SO(2l + 1)/(U(1) × U(p) × SO(2(l - p - 1) + 1)) and SO(2l)/(U(1) × U(p) × SO(2(l - p - 1))) we prove existence of at least two non-Kahler–Einstein metrics. For small values of l and p we give the precise number of invariant Einstein metrics.
TL;DR: In this paper, homothetic vector fields on Randers manifolds of projectively isotropic flag curvature were used to construct new Randers metrics of scalar flag curvatures.
Abstract: We study homothetic vector fields on Randers manifolds of projectively isotropic flag curvature and use them to construct new Randers metrics of scalar flag curvature.
TL;DR: In this paper, the Lebesgue covering lemma is extended from the setting of metric spaces to the set of admissible spaces, a topological space endowed with an admissible family of open coverings, and need not be metric.
Abstract: In this paper the Lebesgue covering lemma is extended from the setting of metric spaces to the setting of admissible spaces. An admissible space is a topological space endowed with an admissible family of open coverings, and need not be metric. The paper contains applications to uniform continuity and dimension theory.
TL;DR: In this article, the additive structure of the integral homology groups of the moduli spaces of semi-stable sheaves on the projective plane having rank and Chern classes (0, 5, 19) was determined using the Bialynicki-Birula method.
Abstract: Using the Bialynicki-Birula method, we determine the additive structure of the integral homology groups of the moduli spaces of semi-stable sheaves on the projective plane having rank and Chern classes (5, 1, 4), (7, 2, 6), respectively, (0, 5, 19). We compute the Hodge numbers of these moduli spaces.
TL;DR: In this article, Pandharipande et al. compute Gromov-Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2, and derive a Caporaso-Harris type formula for counting curves subject to arbitrary additional tangency conditions with respect to the chosen conic.
Abstract: We compute Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 2. The genus zero invariants have been computed a long ago [P. Di Francesco and C. Itzykson, Quantum intersection rings, in The Moduli Space of Curves, eds. R. Dijkgraaf et al., Progress in Mathematics, Vol. 129 (Birkhauser, Boston, 1995), pp. 81–148; L. Gottsche and R. Pandharipande, The quantum cohomology of blow-ups of ℙ2 and enumerative geometry, J. Differential Geom.48(1) (1998) 61–90], Gromov–Witten invariants of any genus for Del Pezzo surfaces of degree ≥ 3 have been found by Vakil [Counting curves on rational surfaces, Manuscripta Math.102 (2000) 53–84]. We solve the problem in two steps: (1) we consider curves on , the plane blown up at one point, which have given degree, genus, and prescribed multiplicities at fixed generic points on a conic that avoids the blown-up point; then we obtain a Caporaso–Harris type formula counting such curves subject to arbitrary additional tangency conditions with respect to the chosen conic; as a result we count curves of any given divisor class and genus on a surface of type , the plane blown up at six points on a given conic and at one more point outside the conic; (2) in the next step, we express the Gromov–Witten invariants of via enumerative invariants of , using Vakil's extension of the Abramovich–Bertram formula.
TL;DR: In this article, up to birational equivalence, the positivity of the adjoint bundles of KX + rL for high rational numbers r is studied, where r is the number of quasi-polarized pairs.
Abstract: Let (X, L) be a quasi-polarized pair, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles KX + rL for high rational numbers r. For this we run a Minimal Model Program with scaling relative to the divisor KX + rL. We give then some applications, namely the classification up to birational equivalence of quasi-polarized pairs with sectional genus 0, 1 and of embedded projective varieties X ⊂ ℙN with degree smaller than 2 codimℙN(X) + 2.
TL;DR: In this article, the authors consider the problem of prescribing pseudo-Hermitian scalar curvature on a compact strictly pseudoconvex CR manifold M and prove that for any negative smooth function f, the pseudo-hermitian curvature can be prescribed to f, provided that dim M = 3 and the CR Yamabe invariant of M is negative.
Abstract: In this paper, we consider the problem of prescribing pseudo-Hermitian scalar curvature on a compact strictly pseudoconvex CR manifold M. Using geometric flow, we prove that for any negative smooth function f we can prescribe the pseudo-Hermitian scalar curvature to be f, provided that dim M = 3 and the CR Yamabe invariant of M is negative. On the other hand, we establish some uniqueness and non-uniqueness results on prescribing pseudo-Hermitian scalar curvature.
TL;DR: In this article, the authors studied the initial boundary value problem for fourth-order wave equations with nonlinear strain and source terms at high energy level and proved that, for certain initial data in the unstable set, the solution with arbitrarily positive initial energy blows up in finite time.
Abstract: In this paper, we study the initial boundary value problem for fourth-order wave equations with nonlinear strain and source terms at high energy level. We prove that, for certain initial data in the unstable set, the solution with arbitrarily positive initial energy blows up in finite time.
TL;DR: In this paper, it was shown that the set of all closed curves of class Cr on the sphere S2 whose geodesic curvatures are constrained to lie in (κ1, κ2), furnished with the Cr topology has n connected components where and contains circles traversed j times (1 ≤ j ≤ n).
Abstract: Let denote the set of all closed curves of class Cr on the sphere S2 whose geodesic curvatures are constrained to lie in (κ1, κ2), furnished with the Cr topology (for some r ≥ 2 and possibly infinite κ1 < κ2). In 1970, J. Li ttle proved that the space of closed curves having positive geodesic curvature has three connected components. Let ρi = arccot κi(i = 1, 2). We show that has n connected components where and contains circles traversed j times (1 ≤ j ≤ n). The component also contains circles traversed (n - 1) + 2k times, and also contains circles traversed n + 2k times, for any k ∈ N. Further, each of (n ≥ 3) is homeomorphic to SO3 × E, where E is the separable Hilbert space. We also obtain a simple characterization of the components in terms of the properties of a curve and prove that is homeomorphic to whenever .
TL;DR: In this article, the authors classify certain contact hypersurfaces in the Riemannian symmetric space SU2,m/S(U2Um), m ≥ 3.
Abstract: In this paper, we classify certain contact hypersurfaces in the Riemannian symmetric space SU2,m/S(U2Um), m ≥ 3.
TL;DR: In this article, the problem of classifying all maximal antipodal sets in the oriented real Grassmann manifold was reduced to that of all maximal subsets satisfying certain conditions in the set consisting of subsets of cardinality k in {1, …, n}.
Abstract: We reduce the problem of classifying all maximal antipodal sets in the oriented real Grassmann manifold to that of classifying all maximal subsets satisfying certain conditions in the set consisting of subsets of cardinality k in {1, …, n}. Using this reduction we classify all maximal antipodal sets in for k ≤ 4. We construct some maximal antipodal subsets for higher k.
TL;DR: In this article, the authors completely classify invariant Hermitian and Kahler structures, together with their paracomplex analogues, on four-dimensional homogeneous pseudo-Riemannian manifolds with nontrivial isotropy subalgebra.
Abstract: We completely classify invariant Hermitian and Kahler structures, together with their paracomplex analogues, on four-dimensional homogeneous pseudo-Riemannian manifolds with nontrivial isotropy subalgebra.
TL;DR: The authors generalize Bertram, Feinberg and Mukai's results to a much broader range of fixed-determinant Brill-Noether loci, including fixed canonical determinants.
Abstract: In the 1990s, Bertram, Feinberg and Mukai examined Brill–Noether loci for vector bundles of rank 2 with fixed canonical determinant, noting that the dimension was always bigger in this case than the naive expectation. We generalize their results to treat a much broader range of fixed-determinant Brill–Noether loci. The main technique is a careful study of symplectic Grassmannians and related concepts.
TL;DR: In this article, the authors provide structure theorems for quadro-quadric Cremona transformations and for extremal varieties 3-covered by twisted cubics by reinterpreting for these objects the algebraic results on the solvability of the radical of rank 3 Jordan algebras.
Abstract: Via the XJC-correspondence proved in [L. Pirio and F. Russo, Extremal varieties 3-rationally connected by cubics, quadro-quadric Cremona transformations and rank 3 Jordan algebras, submitted] we provide some structure theorems for quadro-quadric Cremona transformations and for extremal varieties 3-covered by twisted cubics by reinterpreting for these objects the algebraic results on the solvability of the radical of Jordan algebras. In this way, we can define the semi-simple part and the radical part of a quadro-quadric Cremona transformation, respectively of an extremal variety 3-covered by twisted cubics, and then describe how general objects are constructed from the semi-simple ones, which are completely classified modulo certain equivalences, via suitable null radical extensions.
TL;DR: In this paper, the authors established a much more general statement for a large class of submanifolds satisfying a growth condition at infinity, namely, e-superbiharmonic subsets of a complete Riemannian manifold.
Abstract: Chen famously conjectured that every submanifold of Euclidean space with harmonic mean curvature vector is minimal. In this note, we establish a much more general statement for a large class of submanifolds satisfying a growth condition at infinity. We discuss in particular two popular competing natural interpretations of the conjecture when the Euclidean background space is replaced by an arbitrary Riemannian manifold. Introducing the notion of e-superbiharmonic submanifolds, which contains each of the previous notions as special cases, we prove that e-superbiharmonic submanifolds of a complete Riemannian manifold which satisfy a growth condition at infinity are minimal.