Showing papers in "International Journal of Mathematics in 1997"
TL;DR: The behavior of the Gieseker space of semistable torsion-free sheaves of rank r and fixed c_1, c_2 on a non-singular projective surface as the polarization varies was studied in this article.
Abstract: We study the behavior of the Gieseker space of semistable torsion-free sheaves of rank r and fixed c_1, c_2 on a non-singular projective surface as the polarization varies. It is shown that the ample cone admits a locally finite chamber structure, and that passing a wall adjacent to a pair of chambers has the effect of modifying the moduli space by a (finite) sequence of flips of the type studied by Thaddeus. The key steps are a modification of Simpson's method and the introduction of a "rationally twisted" moduli space. The result is more general but less explicit than the recent work of Ellingsrud-Goettsche (alg-geom/9410005) and Friedman-Qin (alg-geom/9410007).
152 citations
TL;DR: In this paper, the authors construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz-Krieger algebra for topological Markov shifts.
Abstract: We construct and study C*-algebras associated with subshifts in symbolic dynamics as a generalization of Cuntz–Krieger algebras for topological Markov shifts. We prove some universal properties for the C*-algebras and give a criterion for them to be simple and purely infinite. We also present an example of a C*-algebra coming from a subshift which is not conjugate to a Markov shift.
117 citations
TL;DR: In this paper, a description of compact Hermitian complex surfaces whose Riemannian Ricci tensor is of type (1,1) is given. And a complete proof of the Riemmannian version of the Goldberg-Sachs theorem is given in the framework of "classical" Hermitians.
Abstract: The paper contains a description of compact Hermitian complex surfaces whose Riemannian Ricci tensor is of type (1,1). This in turn comes as a consequence of a Riemannian version of the well-known (generalized) Goldberg–Sachs theorem of the General Relativity. A complete proof of the Riemannian version is given in the framework of "classical" Hermitian geometry. The paper includes some more results also pertaining to "Riemannian Goldberg–Sachs theory", as well as a "dual theory" involving the Penrose operator.
99 citations
TL;DR: In this paper, the authors prove some version of Morrison's conjecture on the cone of divisors for Calabi-Yau fiber spaces with non-trivial base pace whose total space is 3-dimensional.
Abstract: We prove some version of Morrison's conjecture on the cone of divisors for Calabi-Yau fiber spaces with non-trivial base pace whose total space is 3-dimensional.
98 citations
TL;DR: In this article, the authors associate to an algebraic quantum group a C*-algebraic quantum groups and show that this C *-Algebraic Quantum Group essentially satisfies an upcoming definition of Masuda, Nakagami and Woronowicz.
Abstract: We associate to an algebraic quantum group a C*-algebraic quantum group and show that this C*-algebraic quantum group essentially satisfies an upcoming definition of Masuda, Nakagami and Woronowicz.
95 citations
TL;DR: In this article, a complex one-parameter family of non-commutative Lp-spaces was constructed by using Calderon's complex interpolation method, and it was shown that all the parametrized Lpspaces are mutually isometrically isomorphic as Banach spaces via natural maps.
Abstract: For a von Neumann algebra ℳ and a weight φ on ℳ, we will construct a complex one-parameter family of non-commutative Lp-spaces by using Calderon's complex interpolation method. This is a simultaneous and complete extension of the construction of non-commutative Lp-spaces by H. Kosaki and M. Terp. Moreover, we will show that for each p, all the parametrized Lp-spaces are mutually isometrically isomorphic as Banach spaces via natural maps.
48 citations
TL;DR: In this paper, the authors give an analogous characterization of Longo's canonical endomorphism in the bimodule theory, and by using this, they construct an inclusion of factors of type II1 from a finite system of Bimodules as a parallel construction to that of Longó-Rehren in a type III setting.
Abstract: We give an analogous characterization of Longo's canonical endomorphism in the bimodule theory, and by using this, we construct an inclusion of factors of type II1 from a finite system of bimodules as a parallel construction to that of Longo–Rehren in a type III setting. When the original factors are approximately finite dimensional, we prove this new inclusion is isomorphic to the asymptotic inclusion in the sense of Ocneanu. This solves a conjecture of Longo–Rehren.
47 citations
TL;DR: In this article, it was proved that the comparability conditions (FCQ), stable (stable) and stable (fcQ+) are equivalent for simple C*-algebras with real rank zero and stable rank one.
Abstract: In this paper we give a representation theorem for the Cuntz monoid S(A) of a σ-unital C*-algebra A with real rank zero and stable rank one, which allows to prove several Riesz decomposition properties on the monoid. As a consequence, it is proved that the comparability conditions (FCQ), stable (FCQ) and (FCQ+) are equivalent for simple C*-algebras with real rank zero. It is also shown that the Grothendieck group of S(A) is a Riesz group, and lattice-ordered under some additional assumptions on A.
45 citations
TL;DR: In this paper, a condition on the outer isomorphisms of Galois groups that preserve the higher ramification groups is discussed, and the condition discussed here is not a natural condition, but at least it gives some sort of idea of how good an inner isomorphism of a Galois group must be to arise geometrically.
Abstract: On the one hand, one knows (cf. the Remark in [4] following Theorem 4.2) that the Grothendieck Conjecture cannot hold in the naive sense (i.e., if one removes the condition of “compatibility with the filtrations” from the outer isomorphisms considered – see, e.g., [8]), so one must put some sort of condition on the outer isomorphisms of Galois groups that one considers. The condition discussed here is that they preserve the higher ramification groups. One can debate how natural a condition this is, but at least it gives some sort of idea of how “good” an outer isomorphism of Galois groups must be in order to arise “geometrically.”
45 citations
TL;DR: In this article, completely bounded multipliers of Kac algebras have been studied in the context of operator space theory, and an appropriate definition for the concept has been proposed.
Abstract: Multipliers, in particular completely bounded multipliers, of Fourier algebras have played a very important role in the study of harmonic analysis and of group von Neumann algebras and C*-algebras. In this paper, we plan to extend this notion to a more general context, i.e. we are going to study the completely bounded multipliers of Kac algebras. We will introduce an appropriate definition for completely bounded multipliers of Kac algebras, study some of basic properties and their connection with the strong Voiculescu amenability of Kac algebras. Operator space theory will play a very important role throughout this paper.
37 citations
TL;DR: In this paper, the existence of very ample vector bundles on curves was studied and it was shown that such bundles can be found over curves of genus at least two, and that they are special cases of a conjecture on scrolls of small codimension.
Abstract: We study very ample vector bundles on curves. We first give numerical conditions for the existence of non-special such bundles. Then we prove the inequality \[ h^0(\det E)\ge h^0(E) + {\rm rank}(E)-2 \] over curves of genus at least two. We apply this to prove some special cases of a conjecture on scrolls of small codimension.
TL;DR: In this paper, it was shown that many results known to hold when is a smooth manifold extend to this more general singular case, and infinite families of examples with which sharply differs from the smooth case, where there is only one such manifold.
Abstract: Any compact 3-Sasakian manifold is a principal circle V-bundle over a compact complex orbifold . This orbifold has a contact Fano structure with a Kahler–Einstein metric of positive scalar curvature and it is the twistor space of a positive compact quaternionic Kahler orbifold . We show that many results known to hold when is a smooth manifold extend to this more general singular case. However, we construct infinite families of examples with which sharply differs from the smooth case, where there is only one such .
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TL;DR: In this article, the Sarkisov program was extended to a logarithmic generalization of the Castelnuovo-Nother Theorem by introducing the notion of the Sarkissov relation, and the mechanism of termination connected to the boundedness conjecture of (log) Q-Fano n-folds was clarified.
Abstract: The purpose of this paper is two-fold. The first is to give a tutorial introduction to the Sarkisov program, a 3-dimensional generalization of Castelnuovo-Nother Theorem ``untwisting" birational maps between Mori fiber spaces, which was recently established by Corti after Sarkisov and Reid.
The second is an attempt to give a logarithmic generalization of the program, introducing the notion of the Sarkisov relation. We give a simple description of the structure of the program and clarify the mechanism of termination connected to the boundedness conjecture of (log) Q-Fano n-folds by Borisov.
TL;DR: In this article, a geometrical condition (PHWC) which is weaker than horizontal weak conformality was studied and it was shown that harmonic maps satisfying this condition include harmonic morphisms and can be described as pulling back certain germs to certain other germs.
Abstract: We study a geometrical condition (PHWC) which is weaker than horizontal weak conformality. In particular, we show that harmonic maps satisfying this condition, which will be called pseudo harmonic morphisms, include harmonic morphisms and can be described as pulling back certain germs to certain other germs. Finally, we construct a canonical f-structure associated to every map satisfying (PHWC) and find conditions on this f-structure to ensure the harmonicity of the map.
TL;DR: In this article, two descriptions of quaternionic Kahler quotients by proper group actions are given: the first as a union of smooth manifolds, some of which come equipped with quaternional or locally Kahler structures; the second as an infinite set of quaterionic kahler spheres.
Abstract: Two descriptions of quaternionic Kahler quotients by proper group actions are given: the first as a union of smooth manifolds, some of which come equipped with quaternionic Kahler or locally Kahler structures; the second as a union of quaternionic Kahler orbifolds. In particular the quotient always has an open set which is a smooth quaternionic Kahler manifold. When the original manifold and the group are compact, we describe a length space structure on the quotient. Similar descriptions of singular hyperKahler and 3-Sasakian quotients are given.
TL;DR: In this paper, the hull-kernel topology of the Haagerup tensor product is investigated and the semi-continuity of norm functions on ideal spaces of C*-algebras A and B is studied.
Abstract: Following the work of Allen, Sinclair and Smith on the primitive ideal space of the Haagerup tensor product A ⊗ hB of C*-algebras A and B, we investigate the hull-kernel topology and use this to determine various other ideal spaces and their topologies in relation to the corresponding ideal spaces of A and B. We study the semi-continuity of norm functions I → ||x + I||(x ∈ A ⊗h B) on these ideal spaces and identify the separated points of Prim(A ⊗h B). Finally, we exhibit several conditions each of which is equivalent to the quasi-standardness of A ⊗h B.
TL;DR: In this article, the authors studied the holonomy and the horizontal holonomy of sub-Riemannian manifolds of contact type relative to an adapted connection, and obtained an Ambrose-Singer type theorem for the horizontal Holonomy.
Abstract: A sub-Riemannian manifold is a smooth manifold which carries a distribution equipped with a metric We study the holonomy and the horizontal holonomy (ie holonomy spanned by loops everywhere tangent to the distribution) of sub-Riemannian manifolds of contact type relative to an adapted connection In particular, we obtain an Ambrose–Singer type theorem for the horizontal holonomy and we classify the holonomy irreducible sub-Riemannian symmetric spaces (ie homogeneous sub-Riemannian manifolds admitting an involutive isometry whose restriction to the distribution is a central symmetry)
TL;DR: In this article, the authors developed a systematic theory of non-abelian Seiberg-Witten equations associated with a Spin^G(4)-structure, where G is a closed subgroup of the unitary group U(V) containing the involution -id_V.
Abstract: The ai of this paper is to develop a systematic theory of non-abelian Seiberg-Witten equations. The equations we introduce and study are associated with a Spin^G(4)-structure, where G is a closed subgroup of the unitary group U(V) containing the involution -id_V
TL;DR: In this paper, the authors studied the geometry and topology of geometrically finite complex hyperbolic manifolds, especially their ends, as well as geometry of their holonomy groups.
Abstract: The aim of this paper is to study geometry and topology of geometrically finite complex hyperbolic manifolds, especially their ends, as well as geometry of their holonomy groups. This study is based on our structural theorem for discrete groups acting on Heisenberg groups, on the fiber bundle structure of Heisenberg manifolds, and on the existence of finite coverings of a geometrically finite manifold such that their parabolic ends have either Abelian or 2-step nilpotent holonomy. We also study an interplay between Kahler geometry of complex hyperbolic n-manifolds and Cauchy–Riemannian geometry of their boundary (2n-1)-manifolds at infinity, and this study is based on homotopy equivalence of manifolds and isomorphism of fundamental groups.
TL;DR: In this article, a very short proof of the Melvin-Morton conjecture relating the colored Jones polynomial and the Alexander polynomials of knots is given, based on explicit evaluation of the corresponding weight systems on primitive elements of the Hopf algebra of chord diagrams.
Abstract: We give a very short proof of the Melvin–Morton conjecture relating the colored Jones polynomial and the Alexander polynomial of knots. The proof is based on explicit evaluation of the corresponding weight systems on primitive elements of the Hopf algebra of chord diagrams which, in turn, follows from simple identities between four-valent tensors on the Lie algebra sl2 and the Lie superalgebra gl(1|1). This shows that the miraculous connection between the Jones and Alexander invariants follows from the similarity (supersymmetry) between sl2 and gl(1|1).
TL;DR: In this paper, the authors use the representation theory of the structure group Spin(n) together with the theory of conformally covariant differential operators to generalize results estimating eigenvalues of the Dirac operator to other tensor-spinor bundles, and to get vanishing theorems for the kernels of first-order differential operators.
Abstract: We use the representation theory of the structure group Spin(n), together with the theory of conformally covariant differential operators, to generalize results estimating eigenvalues of the Dirac operator to other tensor-spinor bundles, and to get vanishing theorems for the kernels of first-order differential operators.
TL;DR: In this paper, the dodecahedral conjecture can be obtained from the distance conjecture of 14 and 15 nonoverlapping unit spheres and from the isoperimetric conjecture of Voronoi faces of unit sphere packings.
Abstract: The dodecahedrad conjecture, posed more than 50 years ago, says that the volume of any Voronoi polyhedron of a unit sphere packing in is at least as large as the volume of a regular dodecahedron of inradius 1. In this paper we show how the dodecahedral conjecture can be obtained from the distance conjecture of 14 and 15 nonoverlapping unit spheres and from the isoperimetric conjecture of Voronoi faces of unit sphere packings.
TL;DR: In this paper, the problem of classifying complex Kähler Finsler manifolds with constant negative holomorphic curvature has been studied and it is shown that up to biholomorphic isometry, there are only three possibilities: C endowed with the euclidean metric, P(C) endowed with (a suitable costant multiple of) the Fubini-Study metric, and the unit ball B in C with respect to the hyperbolic metric.
Abstract: 0. Introduction The classification of simply connected Kähler manifolds of constant holomorphic curvature is a classical result. According to the classification, up to biholomorphic isometry there are only three possibilities: C endowed with the euclidean metric, P(C) endowed with (a suitable costant multiple of) the Fubini-Study metric, and the unit ball B in C endowed with (a suitable costant multiple of) the hyperbolic metric. In recent years questions coming from geometric function theory, and in particular the study of invariant metrics of complex manifolds, suggested to investigate the geometry of complex Finsler (rather than Hermitian) metrics with constant holomorphic curvature, satisfying some natural Kähler condition (agreeing with the usual one in the case of Hermitian metrics) and whose curvature has symmetries enjoyed by the function theoretic examples. In [AP1], and then in [AP2], among other results it was shown that these hypotheses are equivalent to the existence of geodesic complex curves. Since complex Finsler metrics have been considered for quite some time (we recall among other contributions [Ri] who possibly introduced them, [Ru], [K] who indicated the right setting for their study, [Ro], [F], [P]) it is natural to ask whether, at least under natural geometric assumptions, it is possible to obtain a satisfactory classification. Examples show that one should not expect a short list of models. In fact the strongly convex domains in C with their Kobayashi metric provide an infinite dimensional family of not equivalent (neither holomorphically nor isometrically) complex (weakly) Kähler Finsler manifolds of constant negative holomorphic curvature. Furthermore it is easy to endow C with infinite non isometric flat complex Kähler Finsler metrics (the strongly pseudoconvex Minkowski metrics). On the other hand, no example is known of non Hermitian complex Kähler Finsler manifold of positive constant holomorphic curvature. The difference of availability of examples seems to hint that there is a different situation according to the sign of the curvature, in striking contrast with the Hermitian situation. Indeed there are difficulties which do not allow one to extend easily the techniques of the Hermitian case — and even in the real case the classification of constant curvature Finsler manifolds is not clearly established. Finally, the relationship between complex and real geometry is not as effective as in the Hermitian situation. In this paper, using heavily the results of [AP2] and the previous work on the subject by the authors (in particular [AP1]), we address the classification problem and we are able to clarify the situation completely in the non-negative case and make some substantial progress in the negative one. Our work shows that the examples gave the right feeling about the problem. Namely, up to biholomorphic isometries, if some natural symmetry of the curvature is assumed the only complex Kähler Finsler manifold of positive constant holomorphic curvature is P(C) endowed with (a suitable constant multiple of) the Fubini-Study metric, and the only simply connected flat ones are C endowed with strongly pseudoconvex Minkowski metrics. For the negative case we are able to give sufficient conditions ensuring the existence of a Monge-Ampère exhaustion as in the case of strongly convex domains in C and to show that the metric is (a suitable multiple of) the Kobayashi metric of M . We like to thank J. Bland for some very useful remarks and his interest in our work.
TL;DR: In this article, the authors extend the notion of orthogonal multiplication to multilinear norm-preserving mapping, using them to construct new eigenmaps into spheres.
Abstract: We extend the notion of orthogonal multiplication to multilinear norm-preserving mapping, using them to construct new eigenmaps into spheres. We characterize those which are harmonic morphisms. By the method of reduction we construct interesting families of harmonic morphisms into S2 from the product manifolds H2 × S3 and S3 × S3 of hyperbolic spaces and spheres. The corresponding reduction equation depends on two independent variables. We are able to solve the first-order horizontal conformality problem explicitly in terms of elliptic functions and then render the map harmonic by a conformal deformation of the metric.
TL;DR: In this article, the real rank and the stable rank of full group C*-algebras were derived for the free group with n generators, where G1 and G2 are finite groups and |G| is the order of the group G.
Abstract: We compute the real rank and the stable rank of full group C*-algebras. Main result is (i) rr(C*(Fn)) = ∞, (ii) sr(C*(G1 * G2)) = ∞(|G1| ≥ 2, |G2| ≥ 2 and |G1| + |G2| ≥ 5), (iii) sr(C*(G1 * G2)) = 1(|G1| = |G2| = 2), where Fn is the free group with n generators, G1 and G2 are finite groups and |G| means the order of the group G.
TL;DR: In this article, a simply connected, complete, quaternionic Kahler manifold without flat de Rham factor is considered and any 1-parameter group of transformations of M which preserve the quaternion structure Q preserves also the metric g.
Abstract: Let (M,g,Q) be a simply connected, complete, quaternionic Kahler manifold without flat de Rham factor. Then any 1-parameter group of transformations of M which preserve the quaternionic structure Q preserves also the metric g. Moreover, if (M,g) is irreducible then the quaternionic Kahler metric g on (M,Q) is unique up to a homothety.