Semi-free Hamiltonian circle actions on 6-dimensional symplectic manifolds
Transactions of the American Mathematical Society
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 2 (M ) < 3. We give a complete list of the possible manifolds, and determine their equivariant cohomology rings and equivariant Chern classes. Some of these manifolds are classified up to diffeomorphism. We also show the existence for a few
... tence for a few cases. 4543 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4544 HUI LI are (±1, 0), (0, ±1), (±1, ±1). Recall that the coadjoint orbit through an element x of t * is a symplectic manifold and the projection to t * is a moment map for the T 2 -action. The fixed points for the T 2 action are exactly the Weyl group orbit of x in t * . The isotropy weights at a fixed point y are exactly those roots α ∈ t * for which α, y < 0. Now consider the coadjoint orbit of SO(5) through the point (1, 0). The Weyl group orbit of this point consists of the points (1, 0), (−1, 0), (0, 1), and (0, −1). See the following picture for the moment map image of T 2 : The isotropy weights at (1, 0) are (−1, 1), (−1, 0), and (−1, −1). It is easy to see that the e × S 1 action is semi-free and it has 3 fixed point set components: an isolated maximum, an index 2 sphere and an isolated minimum. Let S 1 act on a connected, compact symplectic manifold (M, ω) with moment map φ. Let a ∈ im(φ); then φ −1 (a)/S 1 = M a is called the reduced space at a. If the action is semi-free, then, for any regular value a, M a is a smooth manifold; for a singular value a, it is not clear whether or not M a is a smooth manifold. However, in dimension 6, the fixed point sets are the minimum, the maximum or of index 2 or co-index 2. By [GS], all the reduced spaces (including those on critical levels) are smooth manifolds. Moreover, when the index 2 and co-index 2 fixed point sets consist of surfaces, the reduced spaces are all diffeomorphic, and the index 2 (also co-index 2) surface is symplectically embedded in this reduced space. Now assume the fixed point set only consists of surfaces. Then the reduced spaces are all diffeomorphic to an S 2 bundle over a Riemann surface. (It is enough to consider the reduced space at a regular value right above the minimum Riemann surface.) Moreover, the natural diffeomorphism between the reduced spaces is induced by the Morse flow on M . More explicitly, if there is no critical set between two level sets, the Morse flow gives a one-to-one correspondence between the level sets. Assume there is an index 2 (also co-index 2) fixed surface F with φ(F ) = c. Assume F is the only fixed point set in φ −1 (c − , c + ). Then each S 1 orbit in this neighborhood which flows into F corresponds to a unique point of F and corresponds to a unique S 1 orbit which flows out of F . (See [Mc] for details.) So in both cases, there is a well defined diffeomorphism between the reduced spaces. This diffeomorphism takes a homology class of the reduced space at a regular level right above the minimum to a homology class of the reduced space at a regular level right below the maximum. We give the following definition. Definition 1. Let (M, ω) be a compact connected 6-dimensional symplectic manifold equipped with a semi-free Hamiltonian circle action. Assume the fixed point set consists only of surfaces. If the fibers of the reduced spaces at a regular level right above the minimum and right below the maximum can be represented by the same cohomology class of the reduced space, we say there is no twist; otherwise, we say there is a twist.