A note on Stein fillability of circle bundles over symplectic manifolds

We show that, given a closed integral symplectic manifold (Σ, ω) of dimension 2n ≥ 4, for every integer k>∫Σωⁿ, the Boothby-Wang bundle over (Σ, kω) carries no Stein fillable contact structure. This negatively answers a question raised by Eliashberg. A similar result holds for Boothby-Wang orbibundles. As an application, we prove the non-smoothability of some isolated singularities.