222 research outputs found
Berezin-Toeplitz quantization for compact Kaehler manifolds. A Review of Results
This article is a review on Berezin-Toeplitz operator and Berezin-Toeplitz
deformation quantization for compact quantizable Kaehler manifolds. The basic
objects, concepts, and results are given. This concerns the correct
semi-classical limit behaviour of the operator quantization, the unique
Berezin-Toeplitz deformation quantization (star product), covariant and
contravariant Berezin symbols, and Berezin transform. Other related objects and
constructions are also discussed.Comment: 32 page
Foliation groupoids and their cyclic homology
In this paper we study the Lie groupoids which appear in foliation theory. A
foliation groupoid is a Lie groupoid which integrates a foliation, or,
equivalently, whose anchor map is injective. The first theorem shows that, for
a Lie groupoid G, the following are equivalent:
- G is a foliation groupoid,
- G has discrete isotropy groups,
- G is Morita equivalent to an etale groupoid.
Moreover, we show that among the Lie groupoids integrating a given foliation,
the holonomy and the monodromy groupoids are extreme examples.
The second theorem shows that the cyclic homology of convolution algebras of
foliation groupoids is invariant under Morita equivalence of groupoids, and we
give explicit formulas. Combined with the previous results of Brylinski, Nistor
and the authors, this theorem completes the computation of cyclic homology for
various foliation groupoids, like the (full) holonomy/monodromy groupoid, Lie
groupoids modeling orbifolds, and crossed products by actions of Lie groups
with finite stabilizers. Some parts of the proof, such as the H-unitality of
convolution algebras, apply to general Lie groupoids.
Since one of our motivation is a better understanding of various approaches
to longitudinal index theorems for foliations, we have added a few brief
comments at the end of the second section.Comment: 18 page
Discrete compactness for the hp version of rectangular edge finite elements
International audienceDiscretization of Maxwell eigenvalue problems with edge finite elements involves a simultaneous use of two discrete subspaces of H^1 and H(rot), reproducing the exact sequence condition. Kikuchi's Discrete Compactness Property, along with appropriate approximability conditions, implies the convergence of discrete eigenpairs to the exact ones. In this paper we prove the discrete compactness property for the edge element approximation of Maxwell's eigenpairs on general hp adaptive rectangular meshes. Hanging nodes, yielding 1-irregular meshes, are covered, and the order of the used elements can vary from one rectangle to another, thus allowing for a real hp adaptivity. As a particular case, our analysis covers the convergence result for the p-method
Anabelian geometry and descent obstructions on moduli spaces
We study the section conjecture of anabelian geometry and the sufficiency of
the finite descent obstruction to the Hasse principle for the moduli spaces of
principally polarized abelian varieties and of curves over number fields. For
the former we show that the section conjecture fails and the finite descent
obstruction holds for a general class of adelic points, assuming several
well-known conjectures. This is done by relating the problem to a local-global
principle for Galois representations. For the latter, we prove some partial
results that indicate that the finite descent obstruction suffices. We also
show how this sufficiency implies the same for all hyperbolic curves.Comment: exposition improve
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