Circle and Torus Actions on Equal Symplectic Blow-ups of Cp

A manifold obtained by k simultaneous symplectic blow-ups of CP of equal sizes ǫ (where the size of CP ⊂ CP is one) admits an effective two dimensional torus action if k ≤ 3 and admits an effective circle action if (k−1)ǫ < 1. We show that these bounds are sharp if 1/ǫ is an integer. 1. Toric actions and circle actions in dimension four Hamiltonian torus actions. Let a torus T ∼= (S) act on a compact connected symplectic manifold (M,ω) of dimension 2n by symplectic transformations. The action is Hamiltonian if there exists a moment map, that is, a map Φ: M → t ∼= R such that dΦj = −ι(ξj)ω for all j = 1, . . . , k, where ξ1, . . . , ξk are the vector fields that generate the torus action. If H(M) = 0 then every symplectic torus action is Hamiltonian. By the convexity theorem [At, GS1], the image of the moment map, ∆ := Φ(M), is a convex polytope. By the equivariant Darboux-Weinstein theorem [W], every T -fixed point has a neighborhood U which is equivariantly symplectomorphic to a neighborhood of the origin in C with T acting linearly. The components Φ = 〈Φ, ξ〉, ξ ∈ g, of the moment map are perfect Morse-Bott functions. See [GS1]. From now on we assume that the action of T is effective. The Delzant theorem. If dimT = 1 2 dimM , the triple (M,ω,Φ) is a symplectic toric manifold, and the T -action is called toric. By the Delzant theorem ([De]; also see [LT]), (M,ω,Φ) is determined by ∆ up to an equivariant symplectomorphism preserving Φ. The inverse image under Φ of a vertex of ∆ is a fixed point for the T -action, and the image of a T -fixed point is a vertex of ∆.