Sobolev Regularity of the Bergman Projection on a Smoothly Bounded Stein Domain that is not Hyperconvex

For every $0<r<\frac{1}{2}$, we will construct a flat K\"ahler manifold $M$ and a relatively compact domain with smooth boundary $\Omega\subset M$ that is Stein but not hyperconvex such that the Bergman projection $P$ on $\Omega$ is regular in the $L^2$ Sobolev space $W^s(\Omega)$ for all $0\leq s<r$ but irregular in $W^r(\Omega)$. On these domains, we will also construct $f\in C^\infty(\overline\Omega)$ such that $Pf\notin C^\infty(\overline\Omega)$. We will prove the same result for the invariant Bergman projection on $(2,0)$-forms. These domains are modelled on a construction of Diederich and Ohsawa.