5,473 research outputs found
Semiclassical almost isometry
Let M be a complex projective manifold, and L an Hermitian ample line bundle
on it. A fundamental theorem of Gang Tian, reproved and strengthened by
Zelditch, implies that the Khaeler form of L can be recovered from the
asymptotics of the projective embeddings associated to large tensor powers of
L. More precisely, with the natural choice of metrics the projective embeddings
associated to the full linear series |kL| are asymptotically symplectic, in the
appropriate rescaled sense. In this article, we ask whether and how this result
extends to the semiclassical setting. Specifically, we relate the Weinstein
symplectic structure on a given isodrastic leaf of half-weighted
Bohr-Sommerfeld Lagrangian submanifolds of M to the asymptotics of the the
pull-back of the Fubini-Study form under the semiclassical projective maps
constructed by Borthwick, Paul and Uribe.Comment: exposition improve
Non-commutative integrable systems on -symplectic manifolds
In this paper we study non-commutative integrable systems on -Poisson
manifolds. One important source of examples (and motivation) of such systems
comes from considering non-commutative systems on manifolds with boundary
having the right asymptotics on the boundary. In this paper we describe this
and other examples and we prove an action-angle theorem for non-commutative
integrable systems on a -symplectic manifold in a neighbourhood of a
Liouville torus inside the critical set of the Poisson structure associated to
the -symplectic structure
Legendrian Distributions with Applications to Poincar\'e Series
Let be a compact Kahler manifold and a quantizing holomorphic
Hermitian line bundle. To immersed Lagrangian submanifolds of
satisfying a Bohr-Sommerfeld condition we associate sequences , where is a
holomorphic section of . The terms in each sequence concentrate
on , and a sequence itself has a symbol which is a half-form,
, on . We prove estimates, as , of the norm
squares in terms of . More generally, we show that if and
are two Bohr-Sommerfeld Lagrangian submanifolds intersecting
cleanly, the inner products have an
asymptotic expansion as , the leading coefficient being an integral
over the intersection . Our construction is a
quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of . We prove
that the Poincar\'e series on hyperbolic surfaces are a particular case, and
therefore obtain estimates of their norms and inner products.Comment: 41 pages, LaTe
Special lagrangian fibrations on flag variety
One constructs lagrangian fibrations on the flag variety and proves
that the fibrations are special.Comment: 19 page
Pseudo-Riemannian geodesic foliations by circles
We investigate under which assumptions an orientable pseudo-Riemannian
geodesic foliations by circles is generated by an -action. We construct
examples showing that, contrary to the Riemannian case, it is not always true.
However, we prove that such an action always exists when the foliation does not
contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley's Theorem. As an
application, we show that every Lorentzian surface all of whose
spacelike/timelike geodesics are closed, is finitely covered by .
It follows that every Lorentzian surface contains a non-closed geodesic.Comment: 14 page
Shuffle relations for regularised integrals of symbols
We prove shuffle relations which relate a product of regularised integrals of
classical symbols to regularised nested (Chen) iterated integrals, which hold
if all the symbols involved have non-vanishing residue. This is true in
particular for non-integer order symbols. In general the shuffle relations hold
up to finite parts of corrective terms arising from renormalisation on tensor
products of classical symbols, a procedure adapted from renormalisation
procedures on Feynman diagrams familiar to physicists. We relate the shuffle
relations for regularised integrals of symbols with shuffle relations for
multizeta functions adapting the above constructions to the case of symbols on
the unit circle.Comment: 40 pages,latex. Changes concern sections 4 and 5 : an error in
section 4 has been corrected, and the link between section 5 and the previous
ones has been precise
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