A note on Stein fillability of circle bundles over symplectic manifolds

Abstract

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