TAME CIRCLE ACTIONS SUSAN TOLMAN AND JORDAN WATTS

In this paper, we consider Sjamaar’s holomorphic slice theorem [13], the birational equivalence theorem of Guillemin and Sternberg [7], and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the Kähler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, it does if the fixed points are isolated.) Our main motivation for this paper is that the first author needs the machinery that we develop here to construct a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 fixed points; this answers an open question in symplectic geometry. In this paper, we consider Sjamaar’s holomorphic slice theorem [13], the birational equivalence theorem of Guillemin and Sternberg [7], and a number of important standard constructions that work for Hamiltonian circle actions in both the symplectic category and the Kähler category: reduction, cutting, and blow-up. In each case, we show that the theory extends to Hamiltonian circle actions on complex manifolds with tamed symplectic forms. (At least, it does if the fixed points are isolated.) Our main motivation for this paper is the following question, which appears in McDuff and Salamon [12], and is often referred to as the “McDuff conjecture”: Does there exist a non-Hamiltonian symplectic circle action with isolated fixed points on a closed, connected symplectic manifold? The first author needs the machinery that we develop here to answer this question by constructing a non-Hamiltonian symplectic circle action on a closed, connected six-dimensional symplectic manifold with exactly 32 isolated fixed points in [14]. The results in §7 (and also Proposition 3.1) play a key role in “adding” the isolated fixed points in that paper. Because of this motivation, we focus on the case that the fixed points are isolated, but need to work with a slight generalisation of tamed forms. To explain concretely, we introduce some notation. Let C× act holomorphically on a complex manifold (M,J). Denote by ξM the vector field on M induced by the restricted action of S ⊂ C×. Let MS1 be the set of points fixed by this action, and let Ωk(M)S1 be the set of S-invariant k-forms. Consider a symplectic form ω ∈ Ω2(M)S1 with a moment map Ψ: M → R. Recall that J tames ω if Date: August 21, 2015. Susan Tolman is partially supported by National Science Foundation Grant DMS-1206365. Jordan Watts thanks the University of Illinois at Urbana-Champaign for their support. 1 ω(v, J(v)) > 0 for all non-zero tangent vectors v. In this paper, we will usually work with the weaker condition that ω(ξM , J(ξM)) > 0 on M rMS 1 . Since ω(·, J(·)) is no longer a metric, some of the proofs that work for Kähler manifolds do not work for this larger class without significant modification. However, the following important fact remains true; we will use it repeatedly throught this paper. (Note that, in the Kähler case the gradient flow ∇Ψ is equal to −J(ξM), which is the vector field induced by the R-action given by (t, x) 7→ e · x for all t ∈ R and x ∈M . ) Lemma 1.1. Let (M,J) be a complex manifold with a holomorphic C×-action, a symplectic form ω ∈ Ω2(M)S1 satisfying ω(ξM , J(ξM)) > 0 onMrMS 1, and a moment map Ψ: M → R. Then the function t 7→ Ψ(e · x) is strictly increasing for all x ∈M rMS1. Proof. Given x ∈M rMS1 , let γx(t) := Ψ(e · x). Then for all t ∈ R, γ̇x(t) = dΨ(−J(ξM)) ∣∣∣ et·x = ω(ξM , J(ξM)) ∣∣∣ et·x > 0. 2. Tame Local Normal Form The goal of this section is to prove a C×-equivariant holomorphic Bochner Linearisation Theorem in our tame setting. (More precisely, we develop a C×-equivariant holomorphic local normal form for a neighbourhood of a finite number of fixed points in the same moment fibre.) Our proof is adapted from Sjamaar’s proof of the holomorphic slice theorem for Kähler actions [13, Section 1]. Proposition 2.1. Let (M,J) be a complex manifold with a holomorphic C×-action, a symplectic form ω ∈ Ω2(M)S1 satisfying ω(ξM , J(ξM)) > 0 on M rMS 1, and a moment map Ψ: M → R. Given {p1, . , pk} ∈ MS 1 ∩Ψ−1(0), there exists a C×-invariant neighbourhood of {p1, . , pk} in M which is C×-equivariantly biholomorphic to a neighbourhood of ∐k j=1{0} in ∐k j=1 C, where C× acts linearly on each C. To prove this, we need a holomorphic version of the Bochner Linearisation Theorem for compact group actions on complex manifolds. While this is well-known (cf., for example, [4]), we include the proof for completeness. Since we consider actions on orbifolds in Proposition 7.7, we will prove the orbifold version. Here, if a group G acts on a complex orbifold (M,J), we say that the action respects J if every g ∈ G induces an automorphism of (M,J). Lemma 2.2. Let a compact Lie group G act on a complex orbifold (M,J); assume that the action respects J . Given p ∈ M, there exist a G-equivariant biholomorphism from a G-invariant neighbourhood of 0 ∈ TpM to a neighbourhood of p ∈M . Proof. Let Γ be the (orbifold) isotropy group of p. There exists an extension G̃ of G by Γ, an action of G̃ on an open set Ũ ⊂ C that respects the complex structure and fixes p̃ ∈ Ũ , and a G-equivariant biholomorphism from Ũ/Γ to M that sends [p̃] to p. (See, for example, 2 [11].) Let ψ : Ũ → Tp̃Ũ be any holomorphic map whose differential at p̃ is the identity map on Tp̃Ũ . Since G̃ is compact, we can average ψ to obtain a G̃-equivariant map ψ̄ : Ũ → Tp̃Ũ such that dψ̄|p̃ is equal to the identity, defined by