1. CMB Online first
 Gupta, Purvi

A realanalytic nonpolynomially convex isotropic torus with no attached discs
We show by means of an example in $\mathbb C^3$ that Gromov's
theorem on the presence of attached holomorphic discs for compact
Lagrangian manifolds is not true in the subcritical
realanalytic case, even in the absence of an obvious obstruction,
i.e, polynomial convexity.
Keywords:polynomial hull, isotropic submanifold, holomorphic disc Categories:32V40, 32E20, 53D12 

2. CMB Online first
 Ding, Fan; Geiges, Hansjörg; Zhang, Guangjian

On subcritically Stein fillable 5manifolds
We make some elementary observations concerning subcritically
Stein
fillable contact structures on $5$manifolds.
Specifically, we determine the diffeomorphism type of such
contact manifolds in the case the fundamental group is finite
cyclic,
and we show that on the $5$sphere the standard contact structure
is the unique subcritically fillable one. More generally,
it is shown that subcritically fillable contact structures
on simply connected $5$manifolds are determined by their
underlying almost contact structure. Along the way, we discuss
the
homotopy classification of almost contact structures.
Keywords:subcritically Stein fillable, 5manifold, almost contact structure, thickening Categories:53D35, 32Q28, 57M20, 57Q10, 57R17 

3. CMB 2016 (vol 59 pp. 508)
 De Nicola, Antonio; Yudin, Ivan

Generalized Goldberg Formula
In this paper we prove a useful formula for the graded commutator
of the Hodge
codifferential with the left wedge multiplication by a fixed
$p$form acting on
the de Rham algebra of a Riemannian manifold. Our formula generalizes
a formula
stated by Samuel I. Goldberg for the case of 1forms. As first
examples of
application we obtain new identities on locally conformally KÃ¤hler
manifolds
and quasiSasakian manifolds. Moreover, we prove that under suitable
conditions
a certain subalgebra of differential forms in a compact manifold
is quasiisomorphic as a CDGA to the full de Rham algebra.
Keywords:graded commutator, Hodge codifferential, Hodge laplacian, de Rham cohomology, locally conformal Kaehler manifold, quasiSasakian manifold Categories:53C25, 53D35 

4. CMB 2015 (vol 58 pp. 575)
 MartinezTorres, David

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 4dimensional manifold with a (symplectic) Lie groupoid
structure.
In this short note we show that if $\pi$ is not an area
form on the 2sphere, 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 

5. CMB 2011 (vol 56 pp. 116)
 Krepski, Derek

Central Extensions of Loop Groups and Obstruction to PreQuantization
An explicit construction of a prequantum line bundle for the moduli
space of flat $G$bundles over a Riemann surface is given, where $G$
is any nonsimply 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
prequantization of the moduli space of flat $G$bundles.
Keywords:loop group, central extension, prequantization Categories:53D, 22E 

6. CMB 2011 (vol 54 pp. 716)
 Okassa, Eugène

Symplectic LieRinehartJacobi Algebras and Contact Manifolds
We give a characterization of contact manifolds in terms of symplectic
LieRinehartJacobi algebras. We also give a sufficient condition for a Jacobi
manifold to be a contact manifold.
Keywords:LieRinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifolds Categories:13N05, 53D05, 53D10 

7. CMB 2009 (vol 52 pp. 87)
 Lee, Junho

Holomorphic 2Forms and Vanishing Theorems for GromovWitten Invariants
On a compact K\"{a}hler manifold $X$ with a holomorphic 2form
$\a$, there is an almost complex structure associated with $\a$. We
show how this implies vanishing theorems for the GromovWitten
invariants of $X$. This extends the approach used by Parker and
the author for K\"{a}hler surfaces to higher dimensions.
Category:53D45 

8. CMB 2007 (vol 50 pp. 365)
 Godinho, Leonor

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 

9. CMB 2007 (vol 50 pp. 321)
 Blair, David E.

On Lagrangian Catenoids
Recently I. Castro and F. Urbano introduced the
Lagrangian catenoid.
Topologically, it is $\mathbb R\times S^{n1}$ 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 $(n1)$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 

10. CMB 2005 (vol 48 pp. 561)
 Foth, Philip

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 quasisplit 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 

11. CMB 2003 (vol 46 pp. 277)
 Rochon, Frédéric

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
MorseBott 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 LalondeMcDuffPolterovich
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 

12. CMB 2001 (vol 44 pp. 408)
13. CMB 2001 (vol 44 pp. 36)
 Kapovich, Michael; Millson, John J.

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 sidelengths
$r= (r_1,\dots, r_n)$ 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 
