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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="analytic_geometry">Analytic geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a> (<a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex</a>, <a class="existingWikiWord" href="/nlab/show/rigid+analytic+geometry">rigid</a>, <a class="existingWikiWord" href="/nlab/show/global+analytic+geometry">global</a>)</strong></p> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>+<a class="existingWikiWord" href="/nlab/show/analysis">analysis</a>/<a class="existingWikiWord" href="/nlab/show/analytic+number+theory">analytic number theory</a></p> <h2 id="basic_concepts">Basic concepts</h2> <p><a class="existingWikiWord" href="/nlab/show/analytic+function">analytic function</a></p> <p><a class="existingWikiWord" href="/nlab/show/analytic+space">analytic space</a>, <a class="existingWikiWord" href="/nlab/show/analytic+variety">analytic variety</a>, <a class="existingWikiWord" href="/nlab/show/Berkovich+space">Berkovich space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/polydisc">polydisc</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affinoid+algebra">affinoid algebra</a>, <a class="existingWikiWord" href="/nlab/show/analytic+spectrum">analytic spectrum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+%E2%88%9E-groupoid">analytic ∞-groupoid</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/analytification">analytification</a></p> <h2 id="theorems">Theorems</h2> <p><a class="existingWikiWord" href="/nlab/show/GAGA">GAGA</a></p> </div></div> <h4 id="complex_geometry">Complex geometry</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex+line">complex line</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#InComplexAnalyticGeometry'>In complex analytic geometry</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#general_2'>General</a></li> <li><a href='#in_complex_analytic_geometry_2'>In complex analytic geometry</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <h3 id="general">General</h3> <p>In the context of <a class="existingWikiWord" href="/nlab/show/rigid+analytic+geometry">rigid</a> <a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a>, a <strong>polydisc</strong> is a <a class="existingWikiWord" href="/nlab/show/product">product</a> of <a class="existingWikiWord" href="/nlab/show/discs">discs</a>: the <a class="existingWikiWord" href="/nlab/show/analytic+space">analytic space</a> which is <a class="existingWikiWord" href="/nlab/show/Isbell+duality">formally dual</a> to the <a class="existingWikiWord" href="/nlab/show/Tate+algebra">Tate algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">T_n</annotation></semantics></math> (for an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional polydisk).</p> <p>This is a basic <a class="existingWikiWord" href="/nlab/show/analytic+space">analytic space</a>. It is the analog in <a class="existingWikiWord" href="/nlab/show/analytic+geometry">analytic geometry</a> of the <a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔸</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{A}^n</annotation></semantics></math> in <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>.</p> <p>Those <a class="existingWikiWord" href="/nlab/show/analytic+spaces">analytic spaces</a> which are subspaces of polydiscs are called <em><a class="existingWikiWord" href="/nlab/show/affinoids">affinoids</a></em>.</p> <h3 id="InComplexAnalyticGeometry">In complex analytic geometry</h3> <p>Specifically in <a class="existingWikiWord" href="/nlab/show/complex+analytic+geometry">complex analytic geometry</a> a polydisc is a sub-<a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> of a <a class="existingWikiWord" href="/nlab/show/product">product</a> of <a class="existingWikiWord" href="/nlab/show/complex+planes">complex planes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>↪</mo><msup><mi>ℂ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex"> D \hookrightarrow \mathbb{C}^n </annotation></semantics></math></div> <p>of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><msub><mi>D</mi> <mrow><msub><mi>δ</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>δ</mi> <mi>n</mi></msub></mrow></msub><mo>≔</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>1</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>z</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msup><mi>ℂ</mi> <mi>n</mi></msup><mspace width="thickmathspace"></mspace><mo maxsize="1.2em" minsize="1.2em">|</mo><mspace width="thickmathspace"></mspace><mrow><mo stretchy="false">|</mo><msub><mi>z</mi> <mi>i</mi></msub><mo stretchy="false">|</mo><mo><</mo><msub><mi>δ</mi> <mi>i</mi></msub></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> 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\} </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>i</mi></msub><mo>∈</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>∞</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\delta_i \in (0,\infty]</annotation></semantics></math>.</p> <p>e.g. (<a href="#Maddock">Maddock, p.6</a>)</p> <p>Every <a class="existingWikiWord" href="/nlab/show/complex+analytic+manifold">complex analytic manifold</a> is <a class="existingWikiWord" href="/nlab/show/covering">locally</a> <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphic</a> to a complex polydisc, in that it may be covered by <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> which are <a class="existingWikiWord" href="/nlab/show/biholomorphism">biholomorphic</a> to complex polydiscs. e.g. (<a href="#Maddock">Maddock, p. 7</a>).</p> <p>In fact (<a href="#FornaessStout77a">Fornæss-Stout 77a, lemma II.1</a>) states that every <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a> and <a class="existingWikiWord" href="/nlab/show/second+countable+topological+space">second countable</a> <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a> may already be <a class="existingWikiWord" href="/nlab/show/cover">covered</a> by <a class="existingWikiWord" href="/nlab/show/finite+number">finitely</a> many <a class="existingWikiWord" href="/nlab/show/open+subsets">open subsets</a> <a class="existingWikiWord" href="/nlab/show/biholomorphism">biholomorphic</a> to a polydisc.</p> <p>See also at <em><a href="Stein+manifold#GoodCoversBySteinManifolds">good covers by Stein manifolds</a></em>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/Stein+manifold">Stein manifold</a></li> </ul> <h2 id="references">References</h2> <h3 id="general_2">General</h3> <ul> <li>Leonard Lipshitz, Zachary Robinson, <em>Rings of separated power series</em> (<a class="existingWikiWord" href="/nlab/files/LipshitzRobinson.pdf" title="pdf">pdf</a>)</li> </ul> <h3 id="in_complex_analytic_geometry_2">In complex analytic geometry</h3> <p>Original articles on <a class="existingWikiWord" href="/nlab/show/coverings">coverings</a> of <a class="existingWikiWord" href="/nlab/show/complex+manifolds">complex manifolds</a> by complex polydiscs include</p> <ul> <li id="FornaessStout77a"> <p><a class="existingWikiWord" href="/nlab/show/John+Forn%C3%A6ss">John Fornæss</a>, <a class="existingWikiWord" href="/nlab/show/Edgar+Stout">Edgar Stout</a>, <em>Spreading Polydiscs on Complex Manifolds</em> American Journal of Mathematics Vol. 99, No. 5 (Oct., 1977), pp. 933-960 (<a href="http://www.jstor.org/stable/2373992">JSTOR</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Forn%C3%A6ss">John Fornæss</a>, <a class="existingWikiWord" href="/nlab/show/Edgar+Stout">Edgar Stout</a>, <em>Polydiscs in complex manifolds</em>, Mathematische Annalen 1977, Volume 227, Issue 2, pp 145-153</p> </li> </ul> <p>Introductory lecture notes in the context of the <a class="existingWikiWord" href="/nlab/show/Dolbeault+theorem">Dolbeault theorem</a> include</p> <ul> <li id="Maddock">Zachary Maddock, <em>Dolbeault cohomology</em> (<a href="http://www.math.columbia.edu/~maddockz/notes/dolbeault.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 21, 2023 at 20:51:52. 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