This paper is the sequel to my recent paper [10]. It will provide technical details of our gradient flow construction and related problems, which are essential for our construction of Lagrangian torus fibrations in [10] and subsequent papers [11, 13, 14].

Poisson structures vanishing linearly on a set of smooth closed disjoint curves are generic in the set of all Poisson structures on a compact connected oriented surface. W construct a complete set of invriants classifying these structures up to an orient-preserving Poisson isomorphism. We show that there is a set of non-trivial infinitesimal deformations which generate the second Poisson cohomology and such that each of the deformations changes exactly one of the classifying invarients. As an example, we consider Poisson structures on the sphere which vanish linearly on a set of smooth closed disjoint curves.

In this paper we define invariants of Hamiltonian group actions for central regular values of the moment map. The key hypotheses are that the moment map is proper and that the ambient manifold is symplectically aspherical. The invariants are based on the symplectic vortex equations. Applications include an existence theorem for relative periodic orbits, a computation for circle actions on a complex vector space, and a theorem about the relaton between the invariants introduced here and the Seiberg-Witten invariants of a product of a Riemann surface with a two-sphere.

We describe a simplification of Donaldson's arguments for the construction of symplectic hypersurfaces [4] or Lefschetz pencils [5] that makes it possible to avoid any reference to Yomdin's work on the complexity of real algebraic sets.