Question on region of convergence

Question

We know that the unit circle can be a boundary of complete convergence or complete nonconvergence or of conditional convergence for power series in one complex variable...If the subset of the boundary where there is conditional convergence is P, what can we say about the nature of P for power series in many, but finitely many, complex variables? Can P be a manifold or something wilder? It could be an algebraic geometry question if algebraic geometry were to consider power series, what Euler called infinite polynomials. It seems that the boundary which P could be a subset of, would be in general the product of the boundary of a m-polydisc and the boundary of an n-sphere, depending on which m complex variables are evaluated for convergence separately (leading to boundary of polydisc) and n complex variables together(leading to a hypersphere), where m and n are nonnegative integers adding up to the number of complex variables involved.