Donaldson’s work on Lefschetz pencils has shown that, after a slight perturbation of the symplectic form and a finite number of blow-ups, any closed symplectic manifold (M,ω) can be expressed as a singular fibration with generic fiber a smooth codimension 2 symplectic submanifold. Thus fibrations play a fundamental role in symplectic geometry. It is then natural to study smooth (nonsingular) ru...

the split extension group $a(4)cong 2^7{:}sp_6(2)$ is the affine subgroup of the symplectic group $sp_8(2)$ of index $255$‎. ‎in this paper‎, ‎we use the technique of the fischer-clifford matrices to construct the character table of the inertia group $2^7{:}(2^5{:}s_{6})$ of $a(4)$ of index $63$‎.

Let G be a connected reductive algebraic group over an algebraically closed field k of characteristic zero. We have a left (G×G)-action on G defined as (g1, g2) ·x := g1xg −1 2 . A (G×G)-equivariant embedding G ↪→ X is said to be regular (cf. [BDP], [Br, §1.4]) if the following conditions are satisfied: (i) X is smooth and the complement X \G is a normal crossing divisor D1 ∪ · · · ∪Dn. (ii) Ea...

Given a Lagrangian sphere in a symplectic 4-manifold (M,ω) with b+ = 1, we find embedded symplectic surfaces intersecting it minimally. When the Kodaira dimension κ of (M,ω) is −∞, this minimal intersection property turns out to be very powerful for both the uniqueness and existence problems of Lagrangian spheres. On the uniqueness side, for a symplectic rational manifold and any class which is...

Let M be a closed symplectic manifold, with symplectic form ω. A symplectic automorphism is a diffeomorphism φ : M → M such that φω = ω. We equip the group Aut(M) = Aut(M,ω) of all such maps with the C-topology. Like the whole diffeomorphism group, this is an infinite-dimensional Lie group in a very loose sense: it has a well-defined Lie algebra, which consists of closed oneforms on M , but the...

A n-dimensional Lie group G equipped with a left invariant symplectic form ω+ is called a symplectic Lie group. It is well-known that ω+ induces a left invariant affine structure on G. Relatively to this affine structure we show that the left invariant Poisson tensor π+ corresponding to ω+ is polynomial of degree 1 and any right invariant k-multivector field on G is polynomial of degree at most...