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1. CMB Online first
The Diffeomorphism Type of Canonical Integrations Of Poisson tensors on Surfaces A surface $\Sigma$ endowed with a Poisson tensor
$\pi$ is known to admit
canonical integration, $\mathcal{G}(\pi)$,
which is a 4-dimensional manifold with a (symplectic) Lie groupoid
structure.
In this short note we show that if $\pi$ is not an area
form on the 2-sphere, then $\mathcal{G}(\pi)$ is diffeomorphic
to the cotangent bundle $T^*\Sigma$. This extends
results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini.
Keywords:Poisson tensor, Lie groupoid, cotangent bundle Categories:58H05, 55R10, 53D17 |
2. CMB 2011 (vol 56 pp. 116)
Central Extensions of Loop Groups and Obstruction to Pre-Quantization An explicit construction of a pre-quantum line bundle for the moduli
space of flat $G$-bundles over a Riemann surface is given, where $G$
is any non-simply connected compact simple Lie group. This work helps
to explain a curious coincidence previously observed between
Toledano Laredo's work classifying central extensions of loop groups
$LG$ and the author's previous work on the obstruction to
pre-quantization of the moduli space of flat $G$-bundles.
Keywords:loop group, central extension, prequantization Categories:53D, 22E |
3. CMB 2011 (vol 54 pp. 716)
Symplectic Lie-Rinehart-Jacobi Algebras and Contact Manifolds We give a characterization of contact manifolds in terms of symplectic
Lie-Rinehart-Jacobi algebras. We also give a sufficient condition for a Jacobi
manifold to be a contact manifold.
Keywords:Lie-Rinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifolds Categories:13N05, 53D05, 53D10 |
4. CMB 2009 (vol 52 pp. 87)
Holomorphic 2-Forms and Vanishing Theorems for Gromov--Witten Invariants On a compact K\"{a}hler manifold $X$ with a holomorphic 2-form
$\a$, there is an almost complex structure associated with $\a$. We
show how this implies vanishing theorems for the Gromov--Witten
invariants of $X$. This extends the approach used by Parker and
the author for K\"{a}hler surfaces to higher dimensions.
Category:53D45 |
5. CMB 2007 (vol 50 pp. 365)
Equivariant Cohomology of $S^{1}$-Actions on $4$-Manifolds Let $M$ be a symplectic $4$-dimensional manifold equipped with a
Hamiltonian circle action with isolated fixed points. We describe a
method for computing its integral equivariant cohomology in terms of
fixed point data. We give some examples of these computations.
Categories:53D20, 55N91, 57S15 |
6. CMB 2007 (vol 50 pp. 321)
On Lagrangian Catenoids Recently I. Castro and F. Urbano introduced the
Lagrangian catenoid.
Topologically, it is $\mathbb R\times S^{n-1}$ and its induced metric is
conformally flat,
but not cylindrical. Their result is that if a Lagrangian minimal
submanifold in
${\mathbb C}^n$ is foliated by round $(n-1)$-spheres, it is congruent to
a Lagrangian
catenoid. Here we study the question of conformally flat, minimal, Lagrangian
submanifolds in
${\mathbb C}^n$. The general problem is formidable, but we first show that such a
submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an
eigenvalue of multiplicity one. Then, restricting to the case of at most two
eigenvalues, we show that the submanifold is either flat and totally
geodesic or is
homothetic to (a piece of) the Lagrangian catenoid.
Categories:53C42, 53D12 |
7. CMB 2005 (vol 48 pp. 561)
A Note on Lagrangian Loci of Quotients We study Hamiltonian actions of compact groups in the presence of
compatible involutions. We show that the Lagrangian fixed point set
on the symplectically reduced space is isomorphic to the disjoint
union of the involutively reduced spaces corresponding to
involutions on the group strongly inner to the given one.
Our techniques imply that the solution to the eigenvalues of a sum problem
for a given real form can be reduced to the quasi-split real form in the
same inner class. We also consider invariant quotients with respect to
the corresponding real form of the complexified group.
Keywords:Quotients, involutions, real forms, Lagrangian loci Category:53D20 |
8. CMB 2003 (vol 46 pp. 277)
Rigidity of Hamiltonian Actions This paper studies the following question: Given an
$\omega'$-symplectic action of a Lie group on a manifold $M$ which
coincides, as a smooth action, with a Hamiltonian $\omega$-action,
when is this action a Hamiltonian $\omega'$-action? Using a result of
Morse-Bott theory presented in Section~2, we show in Section~3 of this
paper that such an action is in fact a Hamiltonian $\omega'$-action,
provided that $M$ is compact and that the Lie group is compact and
connected. This result was first proved by Lalonde-McDuff-Polterovich
in 1999 as a consequence of a more general theory that made use of
hard geometric analysis. In this paper, we prove it using classical
methods only.
Categories:53D05, 37J25 |
9. CMB 2001 (vol 44 pp. 408)
Finite Groups Generated by Involutions on Lagrangian Planes of $\mathbf{C}^2$ We classify finite subgroups of $\SO(4)$ generated by anti-unitary
involutions. They correspond to involutions fixing pointwise a
Lagrangian plane. Explicit descriptions of the finite groups and the
configurations of Lagrangian planes are obtained.
Categories:22E40, 53D99 |
10. CMB 2001 (vol 44 pp. 36)
Quantization of Bending Deformations of Polygons In $\mathbb{E}^3$, Hypergeometric Integrals and the Gassner Representation |
Quantization of Bending Deformations of Polygons In $\mathbb{E}^3$, Hypergeometric Integrals and the Gassner Representation The Hamiltonian potentials of the bending deformations of $n$-gons
in $\E^3$ studied in \cite{KM} and \cite{Kl} give rise to a Hamiltonian
action of the Malcev Lie algebra $\p_n$ of the pure braid group
$P_n$ on the moduli space $M_r$ of $n$-gon linkages with the side-lengths
$r= (r_1,\dots, r_n)$ in $\E^3$. If $e\in M_r$ is a singular point we may
linearize the vector fields in $\p_n$ at $e$. This linearization yields a
flat connection $\nabla$ on the space $\C^n_*$ of $n$ distinct points on
$\C$. We show that the monodromy of $\nabla$ is the dual of a quotient
of a specialized reduced Gassner representation.
Categories:53D30, 53D50 |