analytic geometry (complex, rigid, global)
geometry+analysis/analytic number theory
analytic space, analytic variety, Berkovich space
geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
In the context of rigid analytic geometry, a polydisc is a product of discs: the analytic space which is formally dual to the Tate algebra $T_n$ (for an $n$-dimensional polydisk).
This is a basic analytic space. It is the analog in analytic geometry of the affine space $\mathbb{A}^n$ in algebraic geometry.
Those analytic spaces which are subspaces of polydiscs are called affinoids.
Specifically in complex analytic geometry a polydsic is a sub-complex manifold of a product of complex planes
of the form
for $\delta_i \in (0,\infty]$.
e.g. (Maddock, p.6)
Every complex analytic manifold is locally isomorphic to a complex polydisc, in that it may be covered by open subsets which are biholomorphic to complex polydiscs. e.g. (Maddock, p. 7).
In fact (Fornæss-Stout 77a, lemma II.1) states that every connected and second countable complex manifold may already be covered by finitely many open subsets biholomorphic to a polydisc.
See also at good covers by Stein manifolds.
Original articles on coverings of complex manifolds by complex polydiscs include
John Fornæss, Edgar Stout, Spreading Polydiscs on Complex Manifolds American Journal of Mathematics Vol. 99, No. 5 (Oct., 1977), pp. 933-960 (JSTOR)
John Fornæss, Edgar Stout, Polydiscs in complex manifolds, Mathematische Annalen 1977, Volume 227, Issue 2, pp 145-153
Introductory lecture notes in the context of the Dolbeault theorem include