Isolated Fixed Points and Moment Maps on Symplectic Manifolds
نویسنده
چکیده
Let (M, ω) be a compact connected symplectic manifold of dimension 2n equipped with a symplectic circle action. In this paper we show that if the fixed point set is non-empty and isolated then the symplectic circle action must be Hamiltonian. This extends the results of Tolman–Weitsman and McDuff, and proves their conjecture affirmatively. The main strategy is to use a variant of the Euler number at the regular level and its change after passing a critical level.
منابع مشابه
Isolated Fixed Points and Moment Maps of Symplectic Manifolds
Clearly every Hamiltonian circle action on a compact symplectic manifold must have fixed points, due to the existence of the maximum and minimum points of the moment map. The goal of this paper is, conversely, to investigate when a symplectic circle action on a compact symplectic manifold becomes Hamiltonian in terms of the fixed point data. As a consequence, we show that if the fixed point set...
متن کاملCritical Values of Moment Maps on Quantizable Manifolds
Let M be a quantizable symplectic manifold acted on by T = (S) in a Hamiltonian fashion and J a moment map for this action. Suppose that the set M of fixed points is discrete and denote by αpj ∈ Z the weights of the isotropy representation at p. By means of the αpj ’s we define a partition Q+, Q− of M . (When r = 1, Q± will be the set of fixed points such that the half of the Morse index of J a...
متن کاملTowards Generalizing Schubert Calculus in the Symplectic Category
The main purpose of this article is to extend some of the ideas from Schubert calculus to the more general setting of Hamiltonian torus actions on compact symplectic manifolds with isolated fixed points. Given a generic component Ψ of the moment map, which is a Morse function, we define a canonical class αp in the equivariant cohomology of the manifold M for each fixed point p ∈ M. When they ex...
متن کاملSemi-free Hamiltonian Circle Actions on 6 Dimensional Symplectic Manifolds
Assume (M,ω) is a connected, compact 6 dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action, such that the fixed point set consists of isolated points or compact orientable surfaces. We restrict attention to the case dim H(M) < 3. We give a complete list of the possible manifolds, determine their equivariant cohomology ring and equivariant Chern classes. Some of t...
متن کاملSymplectic Toric Manifolds
Foreword These notes cover a short course on symplectic toric manifolds, delivered in six lectures at the summer school on Symplectic Geometry of Integrable Hamiltonian Systems, mostly for graduate students, held at the Centre de Recerca Matemàtica in Barcelona in July of 2001. The goal of this course is to provide a fast elementary introduction to toric manifolds (i.e., smooth toric varieties)...
متن کامل