The Topology of Torus Actions on Symplectic Manifolds

4-manifolds associated b₁ B₂ chapter compact symplectic manifold complex line bundle cones connected Consider convex polyhedron critical point critical submanifold deduce defined definition denotes diffeomorphism dimension endowed equivariant cohomology Euler class exact sequence example exceptional divisor exceptional orbit Exercise fiber fibration Figure finite fixed points function fundamental vector field graph H*BT H²(B hamiltonian action hamiltonian vector fields hermitian homotopy index 2 critical integers invariant symplectic form isomorphism lemma lens space Lie algebra linear matrix moment map morphism normal bundle open subset oriented P¹(C P²(C periodic hamiltonian principal G-bundle Proof Proposition quotient reduced symplectic form regular levels regular values Remark S¹-action S¹-bundle Seifert manifolds Show singular orbits smooth stabilizer subgroup submanifold subspace Supp surface symplectic form symplectic manifold T-action toric variety torus trivial trivialisable tubular neighborhood U₂ vector bundle