One of the remarkable insights of orbifold string theory is an indication of the existence of a new cohomology theory of orbifolds containing the so-called twisted sectors as the contribution of singularities. Mathematically, such an orbifold cohomology theory has been constructed by Chen-Ruan [CR]. The author believes that there is a "stringy" geometry and topology of orbifolds whose core is orbifold cohomology. One aspect of this new geometry and topology is the twisted orbifold cohomology and its relation to discrete torsion. Again, the twisting process has its roots in physics. Physicists usually work over a global quotient X = Y/G only, where G is a finite group acting smoothly on Y. A discrete torsion is a cohomology class α ∈ H²(G, U(1)). Physically, a discrete torsion counts the freedom to choose a phase factor to weight the path integral over each twisted sector without destroying the consistency of string theory. For each α, Vafa-Witten [VW] constructed the twisted orbifold cohomology group H^*_orb,α(X/G,ℂ).

We study torus actions on symplectic manifolds with proper moment maps in the case that each reduced space is two-dimensional. We provide a complete set of invariants for such spaces.

A new technique is presented for construction of Poisson manifolds. This technique is inspired by surgery ideas used to define Poisson structures on 3-manifolds and Gompf's surgery construction for symplectic manifolds. As an application of these ideas it is proved that for all n ≥ d ≥ 4, d even, any finitely presentable group is the fundamental group of a n-dimensional orientable closed Poisson manifold of constant rank d. The unimodularity of some of the Poisson structures thus constructed is studied.

Let M be a compact manifold with a Hamiltonian T action and moment map Φ. The restriction map in rational equivariant cohomology from M to a level set Φ^-1(p) is a surjection, and we denote the kernel by I_(p). When T has isolated fixed points, we show that I_(p) distinguishes the chambers of the moment polytope for M. In particular, counting the number of distinct ideals I_(p) as (p) varies over different chambers is equivalent to counting the number of chambers.

Let M be a connected n-dimensional complex projective manifold and consider an Hermitian ample holomorphic line bundle (L; h_L) on M. Suppose that the unique compatible covariant derivative ▽_L on L has curvature -2πiΩ where Ω is a Kähler form. Let G be a compact connected Lie group and μ: G x M → M a holomorphic Hamiltonian action on (M; Ω ). Let \frac g be the Lie algebra of G, and denote by Φ : M → g^* the moment map.

Let us also assume that the action of G on M linearizes to a holomorphic action on L; given that the action is Hamiltonian, the obstruction for this is of topological nature [GS1]. We may then also assume that the Hermitian structure h_L of L, and consequently the connection as well, are G-invariant. Therefore for every k ∈ N there is an induced linear representation of G on the space H⁰(M;L^⊗k) of global holomorphic sections of L^⊗k. This representation is unitary with respect to the natural Hermitian structure of H⁰(M;L^⊗k) (associated to Ω and h_L in the standard manner). We may thus decompose H⁰(M;L^⊗k) equivariantly according to the irreducible representations of G.

The subject of this paper is the local and global asymptotic behaviour of certain linear series defined in terms this decomposition. Namely, we shall first consider the asymptotic behaviour as k →+ ∞ of the linear subseries of H⁰(M;L^⊗k) associated to a single irreducible representation, and then of the linear subseries associated to a whole ladder of irreducible representations. To this end, we shall estimate the asymptoptic growth, in an appropriate local sense, of these linear series on some loci in M defined in terms of the moment map Φ.