Symplectic resolutions of the quotient of $$ {{\mathbb {R}}}^2 $$ by an infinite symplectic discrete group

Lassoued H.

We give examples of Poisson structures that admit symplectic resolutions of the same dimension. We also give a simple condition under which proper in the smooth case or semi-connected symplectic resolutions in the real analytic and holomorphic case can not exist: open symplectic leaves have to be dense and the singular locus can not be of codimension one.

Guillemin V., Miranda E., Pires A.R.

Let M 2 n be a Poisson manifold with Poisson bivector field Π . We say that M is b -Poisson if the map Π n : M → Λ 2 n ( T M ) intersects the zero section transversally on a codimension one submanifold Z ⊂ M . This paper will be a systematic investigation of such Poisson manifolds. In particular, we will study in detail the structure of ( M , Π ) in the neighborhood of Z and using symplectic techniques define topological invariants which determine the structure up to isomorphism. We also investigate a variant of de Rham theory for these manifolds and its connection with Poisson cohomology.

Bellamy G., Schedler T.

We show that the quotient C 4/G admits a symplectic resolution for $${G = Q_8 \times_{{\bf Z}/2} D_8 < {\sf Sp}_4({\bf C})}$$ . Here Q 8 is the quaternionic group of order eight and D 8 is the dihedral group of order eight, and G is the quotient of their direct product which identifies the nontrivial central elements −Id of each. It is equipped with the tensor product representation $${{\bf C}^2 \boxtimes {\bf C}^2 \cong {\bf C}^4}$$ . This group is also naturally a subgroup of the wreath product group $${Q_8^2 \rtimes S_2 < {\sf Sp}_4({\bf C})}$$ . We compute the singular locus of the family of commutative spherical symplectic reflection algebras deforming C 4/G. We also discuss preliminary investigations on the more general question of classifying linear quotients V/G admitting symplectic resolutions.

Fu B.

Pichereau A.

Resume Nous decrivons la cohomologie de Poisson pour des structures de Poisson sur l'espace affine F 3 , admettant un Casimir quasi-homogene et un lieu singulier reduit a l'origine. Pour citer cet article : A. Pichereau, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

Fu B.

In this paper, firstly we calculate Picard groups of a nilpotent orbit �� in a classical complex simple Lie algebra and discuss the properties of being ℚ-factorial and factorial for the normalization ��tilde; of the closure of ��. Then we consider the problem of symplectic resolutions for ��tilde;. Our main theorem says that for any nilpotent orbit �� in a semi-simple complex Lie algebra, equipped with the Kostant-Kirillov symplectic form ω, if for a resolution π:Z��tilde;, the 2-form π*(ω) defined on π−1(��) extends to a symplectic 2-form on Z, then Z is isomorphic to the cotangent bundle T *(G/P) of a projective homogeneous space, and π is the collapsing of the zero section. It proves a conjecture of Cho-Miyaoka-Shepherd-Barron in this special case. Using this theorem, we determine all varieties ��tilde; which admit such a resolution.

Beauville A.

Karasev M.V.

For general degenerate Poisson brackets, analogues are constructed of invariant vector fields, invariant forms, Haar measure and adjoint representation. A pseudogroup operation is defined that corresponds to nonlinear Poisson brackets, and analogues are obtained for the three classical theorems of Lie. The problem of constructing global pseudogroups is examined. Bibliography: 49 titles.

Lichnerowicz A.