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Made the question a bit more open
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Don Stanley
Don Stanley
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Suppose that $M$ is a symplectic manifold with a Hamiltonian circle action. Is there a topological Lefschetz pencil on $M$, $f\colon M-A \rightarrow S^2$, such that the fibers are symplectic submanifolds of $M$ and such that the circle action restricts to a Hamiltonian circle action on the fibers?

(After reading Tim's answer.)

Are there some cases when there do exist such things?

Suppose that $M$ is a symplectic manifold with a Hamiltonian circle action. Is there a topological Lefschetz pencil on $M$, $f\colon M-A \rightarrow S^2$, such that the fibers are symplectic submanifolds of $M$ and such that the circle action restricts to a Hamiltonian circle action on the fibers?

Suppose that $M$ is a symplectic manifold with a Hamiltonian circle action. Is there a topological Lefschetz pencil on $M$, $f\colon M-A \rightarrow S^2$, such that the fibers are symplectic submanifolds of $M$ and such that the circle action restricts to a Hamiltonian circle action on the fibers?

(After reading Tim's answer.)

Are there some cases when there do exist such things?

Source Link
Don Stanley
Don Stanley
  • 1.4k
  • 11
  • 13

Hamiltonian circle actions and Lefschetz pencils

Suppose that $M$ is a symplectic manifold with a Hamiltonian circle action. Is there a topological Lefschetz pencil on $M$, $f\colon M-A \rightarrow S^2$, such that the fibers are symplectic submanifolds of $M$ and such that the circle action restricts to a Hamiltonian circle action on the fibers?