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August 12, 2011
Abstract
Denote by A θ the rotation algebra corresponding to the rotation 2 πθ . The C*-algebra 𝔹 θ generated by A θ together with certain spectral projections of the canonical unitary generators is studied. The C*-algebra 𝔹 θ is shown to have a unique tracial state and to be nuclear provided that θ is irrational. Moreover, we study the ideal structure of the C*-algebra 𝔹 θ . In particular, it is shown that 𝔹 θ is simple if neither the commutative sub-C*-algebra generated by the spectral projections of u in question (assumed to be a set invariant under Ad v ) nor the corresponding commutative sub-C*-algebra associated to v contains non-zero minimal projections. In the second part of the paper, we study the extended rotation algebra 𝔹 θ generated by the spectral projections (one for each unitary) corresponding to the half-open interval from 0 to θ . (The spectral projections for each half-open interval from nθ to ( n + 1) θ are then included for each integer n .) Using simplicity of 𝔹 θ for θ irrational, the natural field of C*-algebras on the unit circle with fibres 𝔹 θ is shown to be continuous at irrational points. This field is lower semicontinuous on the whole circle. Much more useful is an upper semicontinuous field which is obtained by desingularizing this field at rational points on the circle. The fibres of the desingularized field at rational points are certain (computable) type I C*-algebras. Using this new field, we are able to show that 𝔹 θ is an AF algebra with K 0 (𝔹 θ ) ≅ ℤ + θ ℤ for generic θ , in the sense of Baire category, with the class of the unit being 1 ∈ ℤ.
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June 28, 2011
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The aim of this article is to present a detailed algebraic computation of the Hochschild and cyclic homology groups of the Yang–Mills algebras YM( n ) ( n ∈ ℕ ≧2 ) defined by A. Connes and M. Dubois-Violette in [8], continuing thus the study of these algebras that we have initiated in [17]. The computation involves the use of a spectral sequence associated to the natural filtration on the universal enveloping algebra YM( n ) provided by a Lie ideal 𝔱𝔶𝔪( n ) in 𝔶𝔪( n ) which is free as Lie algebra. As a corollary, we describe the Lie structure of the first Hochschild cohomology group.
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July 14, 2011
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We study the family of smooth rational curves of degree e on a hypersurface of degree d in ℙ n . When e = 1, the smoothness, the dimension and the connectedness of the family have been investigated by W. Barth and A. Van de Ven over the complex number field, and by J. Kollár over an algebraically closed field of arbitrary characteristic. On the other hand, J. Harris, M. Roth, and J. Starr have recently studied irreducibility, smoothness and the dimension of the family over the complex number field. In this paper, we investigate the family in detail, mainly in the case of 1 ≦ e ≦ 3 and d ≧ 1, and in the case of e ≧ 4 and d ≧ 2 e − 3, without the assumption on the characteristic of the ground field.
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July 5, 2011
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This paper focuses on the study of the strong rational connectedness of smooth rationally connected surfaces. In particular, we show that the smooth locus of a log del Pezzo surface is strongly rationally connected. This confirms a conjecture due to Hassett and Tschinkel in [8].
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July 24, 2011
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In this paper we prove a concentration theorem for arithmetic K 0 -theory, this theorem can be viewed as an analog of R. Thomason's result (cf. [22]) in the arithmetic case. We will use this arithmetic concentration theorem to prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. Such a formula was conjectured of a slightly stronger form by K. Köhler and D. Roessler in [16] and they also gave a correct route of its proof there. Nevertheless our new proof is much simpler since it looks more natural and it doesn't involve too many complicated computations.
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June 30, 2011
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We prove that the anti-canonical divisors of weak Fano 3-folds with log canonical singularities are semi-ample. Moreover, we consider semi-ampleness of the anti-log canonical divisor of any weak log Fano pair with log canonical singularities. We show semi-ampleness dose not hold in general by constructing several examples. Based on those examples, we propose sufficient conditions which seem to be the best possible and we prove semi-ampleness under such conditions. In particular we derive semi-ampleness of the anti-canonical divisors of log canonical weak Fano varieties whose lc centers are at most 1-dimensional. We also investigate the Kleiman–Mori cones of weak log Fano pairs with log canonical singularities.
