A note on Stein fillability of circle bundles over symplectic manifolds

We show that, given a closed integral symplectic manifold $(\mathrm{\Sigma },\omega )$ of dimension $2n\ge 4$, for every integer $k>{\int }_{\mathrm{\Sigma }}{\omega }^n$, the Boothby-Wang bundle over $(\mathrm{\Sigma },k\omega )$ 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.