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polydisc

Contents

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 nT_n (for an nn-dimensional polydisk).

This is a basic analytic space. It is the analog in analytic geometry of the affine space 𝔸 n\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 polydsic is a sub-complex manifold of a product of complex planes

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

of the form

D=D δ 1,,δ n{(z 1,,z n) n||z i|<δ i} D = D_{\delta_1, \cdots, \delta_n} \coloneqq \left\{ (z_1, \cdots, z_n) \in \mathbb{C}^n | {\vert z_i\vert \lt \delta_i} \right\}

for δ i(0,]\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.

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)

Revised on June 7, 2014 05:35:49 by Urs Schreiber (89.204.137.10)