Is there a simple argument (or a counterexample) to show that a holomorphically convex subset of an affine algebraic variety is a subvariety which is a compact Riemann surface?



Compact Riemann surfaces as holomorphically convex subsets of affine algebraic varieties
Perhaps I should turn my comment into an "answer".
Affine algebraic varieties over $\mathbb{C}$ are Stein spaces. That is,
they are already holomorphically convex, and points can be separated by global
holomorphic functions. The latter property implies that affine varieties
can never contain compact Riemann surfaces.


