nLab polydisc

Idea

General

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.

In complex analytic geometry

Specifically in complex analytic geometry a polydisc is a sub-complex manifold of a product of complex planes

$$D \hookrightarrow \mathbb{C}^n$$

of the form

$$D = D_{\delta_1, \cdots, \delta_n} \coloneqq \left\{ (z_1, \cdots, z_n) \in \mathbb{C}^n \;\big|\; {\vert z_i\vert \lt \delta_i} \right\}$$

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.

Related concepts

• Stein manifold

References

General

• Leonard Lipshitz, Zachary Robinson, Rings of separated power series (pdf)

In complex analytic geometry

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

• Zachary Maddock, Dolbeault cohomology (pdf)