On the isomorphism question for complete Pick multiplier algebras

Every multiplier algebra of an irreducible complete Pick kernel arises as the restriction algebra $\text{mv}=\{f{\mid }_V:f\in {\text{cM}}_d\}$, where $d$ is some integer or $\mathrm{\infty }$, ${\text{cM}}_d$ is the multiplier algebra of the Drury-Arveson space $H_d^2$, and $V$ is a subvariety of the unit ball. For finite $d$ it is known that, under mild assumptions, every isomorphism between two such algebras $\text{mv}$ and $\text{mw}$ is induced by a biholomorphism between $W$ and $V$. In this paper we consider the converse, and obtain positive results in two directions. The first deals with the case where $V$ is the proper image of a finite Riemann surface. The second deals with the case where $V$ is a disjoint union of varieties.