A counterexample to the differentiability of the Bergman kernel function

In this paper we prove the following main result. Let $D$ be a smoothly bounded pseudoconvex domain in $\mathbb {C}^n$ with $n\ge 2$. Suppose that there exists a complex variety sitting in the boundary $bD$; then we have \[ K_{D}(z,w)\notin C^{\infty }(\overline {D}\times \overline {D}-\Delta (bD)). \] In particular, the Bergman kernel function associated with the Diederich-Fornaess worm domain is not smooth up to the boundary in joint variables off the diagonal of the boundary.