Group Of Symplectic Automorphisms Research Articles

Gerbes, 2-Gerbes and Symplectic Fibrations

Let (F,u)\to P\to N be a symplectic fibration in math.SG/0503268 McDuff has defined a subgroup Ham^s(F,u) of the group of symplectic automorphisms of(F,u). She has shown that the cohomology class [u] of u can be extended to P if and only if the symplectic fibration has an Ham^s reduction. To show this result, she constructs a class who represents the obstruction to extend u. This class can be identified to a 3-class of P using a spectral sequence. The purpose of this paper is to define a 2-gerbe whose classifying cocycle is the class defined by McDuff. To define this 2-gerbe, we construct fundamental gerbes in Dirac geometry which represents the obstruction of [u] to be exact or integral. Using this gerbes we propose a quantization of symplectic manifolds

The Torelli geometry and its applications

Let S be a closed orientable surface of genus g. The mapping class group Mod(S) of S is defined as the group of isotopy classes of orientationpreserving diffeomorphisms S → S. We will need also the extended mapping class group Mod±(S) of S which is defined as the group of isotopy classes of all diffeomorphisms S → S. Let us fix an orientation of S. Then the algebraic intersection number provides a nondegenerate, skew-symmetric, bilinear form on H = H1(S,Z), called the intersection form. The natural action of Mod(S) on H preserves the intersection form. If we fix a symplectic basis in H, then we can identify the group of symplectic automorphisms of H with the integral symplectic group Sp(2g,Z) and the action of ModS on H leads to a natural surjective homomorphism

Analogue of Weil representation for abelian schemes

In this paper we construct a projective action of certain arithmetic group on the derived category of coherent sheaves on an abelian scheme $A$, which is analogous to Weil representation of the group. More precisely, the arithmetic group in question is a congruence subgroup in the group of symplectic automorphisms of $A\times\hat{A}$ where $\hat{A}$ is the dual abelian scheme. The projectivity of this action refers to shifts in the derived category and tensorings with line bundles pulled from the base. In particular, if $A$ is an abelian scheme over $S$ equipped with an ample line bundle $L$ of degree 1 then we construct an action of a central extension of $Sp_{2n}(\Bbb Z)$ by $\Bbb Z\times Pic(S)$ on the derived category of coherent sheaves on $A^n$ (the $n$-th fibered power of $A$ over $S$). We describe the corresponding central extension explicitly using the the canonical torsion line bundle on $S$ associated with $L$. As a main technical result we prove the existence of a representation of rank $d$ for a symmetric finite Heisenberg group scheme of odd order $d^2$.

On symplectic quotients of K3 surfaces

In this note, we construct generalized Shioda-Inose structures on K3 surfaces using cyclic covers and almost functoriality of Shioda-Inose structures with respect to normal subgroups of a given group of symplectic automorphisms.

Generalized Shioda-Inose structures on K3 surfaces

In this note, we study the action of finite groups of symplectic automorphisms on K3 surfaces which yield quotients birational to generalized Kummer surfaces. For each possible group, we determine the Picard number of the K3 surface admitting such an action and for singular K3 surfaces we show the uniqueness of the associated abelian surface.

Symplectic homogeneous spaces

In this paper we make various remarks, mostly of a computational nature, concerning a symplectic manifold X on which a Lie group G acts as a transitive group of symplectic automorphisms. The study of such manifolds was initiated by Kostant [41 and Souriau [5] and was recently developed from a more general point of view by Chu [2]. The first part of this paper is devoted to reviewing the Kostant, Souriau, Chu results and deriving from them a generalization of the Cartan conjugacy theorem. In the second part of this paper we apply these results to Lie algebras admitting a generalized (k, p) decomposition. In this paper we make various remarks, mostly of a computational nature, concerning a symplectic manifold X on which a Lie group G acts as a transitive group of symplectic automorphisms. The study of such manifolds was initiated by Kostant [4] and Souriau [5] and was recently developed from a more general point of view by Chu [2]. For the convenience of the reader we will begin by summarizing the basic facts. 1. General facts. Let G be a Lie group and X = G/H a homogeneous space for G where H is a closed subgroup, and let ir: G G/H = X be the projection. If Q is an invariant form on X then it is clear that a = 7T*Q is a left invariant form on G which satisfies (i) tI a = 0 for all t E h where h is the Lie algebra of H; (ii) a is invariant under right multiplication by elements of H, and hence under Ad for elements of H. Conversely, it is clear that any left invariant form a on G satisfying (i) and (ii) arises from G/H. If Q2 is a symplectic form then it is clear that a left invariant vector field will satisfy tla = 0 if and only if t E h. Furthermore, since do = 0, the set of all vector fields satisfying tla = 0 forms an integrable subbundle of TG, and in particular, the left invariant ones form a subalgebra of the Lie algebra of G; let us call it ha. We have thus recovered h. Let Ha be the group generated by ha. Notice that for any t E ha we have Dto = Ida + d(QJ1) = 0 so that a is invariant under Ha. The only problem is that Ha need not be closed. Let Received by the editors May 23, 1974. AMS (MOS) subject classifications (1970). Primary 53C15, 53C30. (1)This research was partially supported by NSF Grant GP43613X. Copyriglht