New Results on the Bergman Kernel of the Worm Domain in Complex Space

In 1977, Diederich and Fornæss [DIF] constructed a counterexample to the longstanding conjecture that (the closure of) a smoothly bounded, pseudoconvex domain in complex space is the decreasing intersection of smooth, pseudoconvex domains. The smoothly bounded domain Ωβ that they constructed has become known as the “worm”. The worm is smoothly bounded and pseudoconvex. In fact it is a counterexample to several important questions: • The worm Ωβ is not the decreasing intersection of smooth, pseudoconvex domains. • There is a function f that is C∞ on Ωβ, holomorphic on Ωβ, and such that f cannot be approximated uniformly on Ωβ by functions holomorphic on a neighborhood of Ωβ. • The domain Ωβ does not have a global plurisubharmonic defining function. The general concept of the worm domain has several concrete realizations. Two of these that will be important for us are: (i) The unbounded, non-smooth worm