  We study the isomorphism problem for the multiplier algebras of irreducible
complete Pick kernels. These are precisely the restrictions $\mathcal M_V$ of
the multiplier algebra $\mathcal M$ of Drury-Arveson space to a holomorphic
subvariety $V$ of the unit ball $\mathbb{B}_d$.
  We find that $\mathcal M_V$ is completely isometrically isomorphic to
$\mathcal M_W$ if and only if $W$ is the image of $V$ under a biholomorphic
automorphism of the ball. In this case, the isomorphism is unitarily
implemented. This is then strengthend to show that, when $d<\infty$, every
isometric isomorphism is completely isometric.
  The problem of characterizing when two such algebras are (algebraically)
isomorphic is also studied. When $V$ and $W$ are each a finite union of
irreducible varieties and a discrete variety in $\mathbb{B}_d$ with $d<\infty$,
then an isomorphism between $\mathcal M_V$ and $\mathcal M_W$ determines a
biholomorphism (with multiplier coordinates) between the varieties; and the
isomorphism is composition with this function. These maps are automatically
weak-$*$ continuous.
  We present a number of examples showing that the converse fails in several
ways. We discuss several special cases in which the converse does
hold---particularly, smooth curves and Blaschke sequences.
  We also discuss the norm closed algebras associated to a variety, and point
out some of the differences.
