Exact fillability of Boothby–Wang bundles for large k

Determine whether, for any closed integral symplectic manifold (E, ω) of dimension 2n ≥ 4 and any integer k exceeding the integer pairing ⟨[ω]^n, [E]⟩, the Boothby–Wang circle bundle over (E, kω) admits an exactly fillable contact structure.

Background

The paper proves that for any closed integral symplectic manifold (E, ω) of dimension 2n ≥ 4 and any integer k greater than ⟨[ω]^n, [E]⟩, the associated Boothby–Wang bundle over (E, kω) carries no Stein fillable contact structure. Since every Boothby–Wang manifold is strongly fillable, this result leaves open the status of exact fillability, which lies between strong and Stein fillability.

Motivated by prior work and conjectures on exact fillability—for instance, a conjecture by Courte concerning real projective spaces and subsequent results of Zhou ruling out exact fillings for certain Boothby–Wang manifolds—the authors pose a general question about exact fillability for Boothby–Wang bundles over (E, kω) when k is large in the sense k > ⟨[ω]^n, [E]⟩.