Exact fillability of Boothby–Wang bundles for large k

Determine whether, for a closed integral symplectic manifold (Σ, ω) of dimension 2n ≥ 4 and any integer k strictly greater than the integer obtained by evaluating the nth power of [ω] on the fundamental class [Σ], the principal circle bundle over Σ with Euler class −k[ω] (the Boothby–Wang bundle over (Σ, kω)) admits a contact structure that is exactly symplectically fillable.

Background

Boothby–Wang bundles are principal circle bundles over closed integral symplectic manifolds whose natural connection 1-forms define contact structures. While every Boothby–Wang contact manifold is strongly fillable, their Stein fillability is more restrictive. The main theorem establishes that when k exceeds the top-degree pairing of [ω]^n with [Σ], the Boothby–Wang bundle over (Σ, kω) carries no Stein fillable contact structure.

Motivated by recent progress on non-exact fillability for certain Boothby–Wang manifolds, including real projective spaces, the authors raise the broader problem of determining exact fillability in the large-k regime. This shifts focus from Stein fillability to exact symplectic fillability under the same numerical threshold on k.