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 Kähler manifold $M$ and a relatively compact domain with smooth boundary $\mathrm{\Omega }\subset M$ that is Stein but not hyperconvex such that the Bergman projection $P$ on $\mathrm{\Omega }$ is regular in the $L^2$ Sobolev space $W^s(\mathrm{\Omega })$ for all $0\le s<r$ but irregular in $W^r(\mathrm{\Omega })$. On these domains, we will also construct $f\in C^{\mathrm{\infty }}(\bar{\mathrm{\Omega }})$ such that $Pf\mathrm{\notin }{C}^{\mathrm{\infty }}(\bar{\mathrm{\Omega }})$. 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.