A note on Stein fillability of circle bundles over symplectic manifolds

TL;DR

The paper proves that certain circle bundles over symplectic manifolds do not admit Stein fillable contact structures when the bundle's twisting exceeds the symplectic volume, answering a question by Eliashberg.

Contribution

It establishes a negative criterion for Stein fillability of Boothby-Wang bundles over symplectic manifolds, extending understanding of contact topology.

Findings

No Stein fillable contact structures for high twisting integers

Negative answer to Eliashberg's question on Stein fillability

Application to non-smoothability of certain singularities

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.