Hamiltonian Circle Action

• Hamiltonian circle actions are smooth, symplectic S¹ actions defined via a moment map that bridges differential topology and symplectic geometry.
• They are characterized by distinctive fixed point data, isotropy weights, and reduced spaces whose topology (e.g., Betti numbers) distinguishes Hamiltonian from non-Hamiltonian cases.
• Applications include manifold classification, constructive gluing methods, and computation of invariants such as Chern numbers and Duistermaat–Heckman measures.

A Hamiltonian circle action is a smooth, symplectic action of the circle group $S^1$ on a symplectic manifold $(M,\omega)$ for which there exists an $S^1$-invariant function $H:M\to\mathbb{R}$ satisfying $dH = \iota_X\omega$ with $X$ the vector field generated by the action. These actions display deep connections between global symplectic topology, equivariant cohomology, and transformation group theory, serving as a unifying framework for diverse phenomena in geometry and topology.

1. Definition, Moment Maps, and Semifree Actions

A symplectic action of $S^1$ on $(M,\omega)$ is Hamiltonian if there exists a (global) moment map $H:M\to\mathbb{R}$ such that $\iota_X\omega = dH$, where $X$ generates the circle action. In situations where the cohomology class $[\omega]$ is integral, one may initially only have a generalized moment map $p:M\to S^1$ satisfying $dp = \iota_X\omega$, which can be lifted to a genuine Hamiltonian if and only if certain topological obstructions vanish.

A circle action is called semifree if it is free away from the fixed point set (the stabilizer is $S^1$ at fixed points and trivial otherwise). In many classification problems, particularly for 6-dimensional manifolds, semifree Hamiltonian actions are tractable and the fixed point components are typically surfaces (possibly non-isolated) (Cho et al., 2010).

2. Topological and Cohomological Classification Criteria

A fundamental insight (Cho et al., 2010) is the role of the topology of the symplectic reduced spaces $M_{\text{red}} = p^{-1}(t)/S^1$ in determining whether a symplectic $S^1$-action is Hamiltonian. Specifically, for a closed, 6-dimensional symplectic manifold $(M,\omega)$ with $\dim M^{S^1}\leq 2$ and non-empty fixed point set, the action is Hamiltonian if and only if every reduced space $M_{\text{red}}$ has second Betti number $b_2=1$. Non-Hamiltonian actions necessarily force $b_2(M_{\text{red}})\geq 2$. This dichotomy arises from properties of the Duistermaat–Heckman function for the reduced symplectic forms, log-concavity phenomena, and the classification of ruled four-manifolds as symplectic reductions.

When the action is semifree with only surfaces as fixed components, all reduced spaces are diffeomorphic to $S^2$-bundles over a Riemann surface, and the number $k$ of fixed surfaces with positive genus satisfies $k\leq 4$ ($k\neq 2$); when the extremal fixed surfaces are spheres, $k\leq 1$ is enforced (Cho et al., 2010).

3. Fixed Point Data, Weights, and Equivariant Topology

Hamiltonian circle actions exhibit intricate fixed point data. At each isolated fixed point, the linearized representation decomposes into complex 1-dimensional $S^1$-modules (weights), and these weights—subject to global invariants such as Chern classes—encode much of the action's geometry. For actions with isolated fixed points, the structure of the fixed set, together with the isotropy weights, provide powerful constraints:

• For a closed $2n$-dimensional symplectic $S^1$-manifold with the minimal number $n+1$ of isolated fixed points, a single key weight at a fixed point determines the first Chern class $c_1(M)$, the cohomology ring, and the entire set of weights (Li, 2014).
• In higher dimensions (e.g., $8$-dimensional, with $5$ fixed points), the positivity condition on weights implied by the structure of labeled multigraphs ensures the cohomology ring and Chern classes of $M$ agree with those of the complex projective space $\mathbb{CP}^4$ (Jang et al., 2014).
• The equivariant cohomology and localization formulas (Atiyah–Bott, Berline–Vergne) reduce global invariants like Chern numbers to fixed point contributions, thus translating topological and geometric data into local, combinatorial constraints (Godinho et al., 2012).

A salient feature is that rigidity results and classification theorems frequently hinge on properties such as the sums of weights across fixed points, and sharp results can be obtained by analyzing linear relations among them.

4. Moment Maps, Duistermaat–Heckman Theory, and Reduced Spaces

For Hamiltonian circle actions, the behavior of the moment map, symplectic reductions, and the variation of the induced symplectic forms are central. The Duistermaat–Heckman theorem asserts that the cohomology class of the symplectic form on the reduced space varies linearly as

$[\omega_t] = [\omega_r] - e\cdot (t-r)$

with $e$ the Euler class of the principal $S^1$-bundle. At critical levels of the moment map, the Euler class "jumps" by the sum of the Poincaré duals to the fixed point components,

$e_{s+} = e_{s-} + \sum_i PD([X_i])$

for fixed surfaces $X_i$. The adjunction formula

$[X]^2 - c_1(M_{\text{red}})\cdot [X] = 2g(X) - 2$

relates intersection properties of fixed submanifolds to their genus, leading to quantitative restrictions on the fixed set. The log-concavity of the Duistermaat–Heckman function is a key analytic tool; in the periodic case, the function must be constant unless jumps from fixed-point contributions arise, thereby distinguishing Hamiltonian actions from symplectic (but non-Hamiltonian) ones.

5. Concrete Constructions and Examples

The construction of manifolds with prescribed Hamiltonian circle actions is often carried out by gluing together "local pieces," each modeled on neighborhoods of critical levels or regular intervals for the moment map (Cho et al., 2010). Regular pieces correspond to ruled surfaces with prescribed principal $S^1$-bundles and symplectic forms; critical pieces are constructed as complex plane bundles over surfaces or via Guillemin–Sternberg models. Rigidity theorems ensure the boundaries of such pieces can be matched, producing global 6-manifolds with semifree Hamiltonian actions and specified fixed-point data. As demonstrated, examples exist with $k=0,1,3,4$ positive genus surfaces in the fixed set (but never $k=2$), illustrating the sharpness of theoretical constraints. Furthermore, with fixed surfaces of genus zero only, arbitrarily many fixed spheres may arise.

6. Applications and Classification Impact

Hamiltonian circle actions serve as powerful classification tools in symplectic geometry, furnishing invariants for distinguishing Hamiltonian versus non-Hamiltonian cases (e.g., via $b_2(M_{\text{red}})$), as well as determining global topology from local action data. Restrictions on fixed-point structure tie directly to rigidity phenomena, as in the solution to variations of the Petrie conjecture. The interplay between fixed-point geometry, moment map behavior, Duistermaat–Heckman theory, and quantum invariants (e.g., Seidel representation, Gromov–Witten invariants) situates Hamiltonian circle actions at the intersection of transformation group theory and symplectic topology. Methods developed in these studies—notably, gluing constructions, log-concavity arguments, and local-to-global symplectic geometry—have become foundational in equivariant symplectic topology and inform ongoing classifications in both lower and higher-dimensional settings.

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In summary, the theory of Hamiltonian circle actions integrates equivariant topology, symplectic geometry, and transformation group theory, providing both robust classification criteria (e.g., via Betti numbers of reduced spaces) and constructive methods for realizing symplectic manifolds with prescribed symmetry and topology. The foundational work on 6-dimensional semifree cases with non-isolated fixed sets clarifies the limits and mechanisms of such actions, illustrating both the power and delicacy of symplectic and equivariant constraints (Cho et al., 2010).