Topology of symplectic torus actions with symplectic orbits

We give a concise overview of the classification theory of symplectic manifolds equipped with torus actions for which the orbits are symplectic (this is equivalent to the existence of a symplectic principal orbit), and apply this theory to study the structure of the leaf space induced by the action. In particular we show that if M is a symplectic manifold on which a torus T acts effectively with symplectic orbits, then the leaf space M / T is a very good orbifold with first Betti number b 1 ( M / T )=b 1 ( M )−dim T .