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Riemann surface in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/17293/#Item_1" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; 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="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> <h4 id="differential_geometry">Differential geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic</a> <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a></strong></p> <p><strong>Introductions</strong></p> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+1">from point-set topology to differentiable manifolds</a></p> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a>: <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+coordinate+systems">coordinate systems</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+spaces">smooth spaces</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+manifolds+and+orbifolds">manifolds</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+smooth+homotopy+types">smooth homotopy types</a>, <a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">supergeometry</a></p> <p><strong>Differentials</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiation">differentiation</a>, <a class="existingWikiWord" href="/nlab/show/chain+rule">chain rule</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+function">differentiable function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal space</a>, <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+point">infinitesimally thickened point</a>, <a class="existingWikiWord" href="/nlab/show/amazing+right+adjoint">amazing right adjoint</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/V-manifolds">V-manifolds</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>, <a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/analytic+manifold">analytic manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+manifold">formal smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold+structure+of+mapping+spaces">manifold structure of mapping spaces</a></p> </li> </ul> <p><strong>Tangency</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a>, <a class="existingWikiWord" href="/nlab/show/frame+bundle">frame bundle</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+field">vector field</a>, <a class="existingWikiWord" href="/nlab/show/multivector+field">multivector field</a>, <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+forms+in+synthetic+differential+geometry">differential forms</a>, <a class="existingWikiWord" href="/nlab/show/de+Rham+complex">de Rham complex</a>, <a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/pullback+of+differential+forms">pullback of differential forms</a>, <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant differential form</a>, <a class="existingWikiWord" href="/nlab/show/Maurer-Cartan+form">Maurer-Cartan form</a>, <a class="existingWikiWord" href="/nlab/show/horizontal+differential+form">horizontal differential form</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cogerm+differential+form">cogerm differential form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+of+differential+forms">integration of differential forms</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+diffeomorphism">local diffeomorphism</a>, <a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+morphism">formally étale morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+morphism">formally smooth morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/immersion">immersion</a>, <a class="existingWikiWord" href="/nlab/show/formally+unramified+morphism">formally unramified morphism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham space</a>, <a class="existingWikiWord" href="/nlab/show/crystal">crystal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+disk+bundle">infinitesimal disk bundle</a></p> </li> </ul> <p><strong>The magic algebraic facts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivations+of+smooth+functions+are+vector+fields">derivations of smooth functions are vector fields</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hadamard+lemma">Hadamard lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel%27s+theorem">Borel's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Boman%27s+theorem">Boman's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+extension+theorem">Whitney extension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Steenrod-Wockel+approximation+theorem">Steenrod-Wockel approximation theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincare+lemma">Poincare lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Stokes+theorem">Stokes theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/de+Rham+theorem">de Rham theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Kostant-Rosenberg+theorem">Hochschild-Kostant-Rosenberg theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology+hexagon">differential cohomology hexagon</a></p> </li> </ul> <p><strong>Axiomatics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kock-Lawvere+axiom">Kock-Lawvere axiom</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a>, <a class="existingWikiWord" href="/nlab/show/super+smooth+topos">super smooth topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/microlinear+space">microlinear space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+axiom">integration axiom</a></p> </li> </ul> </li> </ul> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">ʃ</mo><mo>⊣</mo><mo>♭</mo><mo>⊣</mo><mo>♯</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\esh \dashv \flat \dashv \sharp )</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/discrete+object">discrete object</a>, <a class="existingWikiWord" href="/nlab/show/codiscrete+object">codiscrete object</a>, <a class="existingWikiWord" href="/nlab/show/concrete+object">concrete object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/points-to-pieces+transform">points-to-pieces transform</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+%28infinity%2C1%29-topos+--+structures">structures in cohesion</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dR-shape+modality">dR-shape modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/dR-flat+modality">dR-flat modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="0em" rspace="thinmathspace">ʃ</mo> <mi>dR</mi></msub><mo>⊣</mo><msub><mo>♭</mo> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">\esh_{dR} \dashv \flat_{dR}</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/infinitesimal+cohesion">infinitesimal cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+modality">classical modality</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/tangent+cohesive+%28%E2%88%9E%2C1%29-topos">tangent cohesion</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+cohomology+diagram">differential cohomology diagram</a></li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a></strong></p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/reduction+modality">reduction modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+shape+modality">infinitesimal shape modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/infinitesimal+flat+modality">infinitesimal flat modality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℜ</mi><mo>⊣</mo><mi>ℑ</mi><mo>⊣</mo><mi>&</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Re \dashv \Im \dashv \&)</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reduced+object">reduced object</a>, <a class="existingWikiWord" href="/nlab/show/coreduced+object">coreduced object</a>, <a class="existingWikiWord" href="/nlab/show/formally+smooth+object">formally smooth object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formally+%C3%A9tale+map">formally étale map</a></p> </li> <li> <p><a href="cohesive+%28infinity%2C1%29-topos+--+infinitesimal+cohesion#StructuresInDifferentialCohesion">structures in differential cohesion</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/super+smooth+infinity-groupoid">graded differential cohesion</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fermionic+modality">fermionic modality</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/bosonic+modality">bosonic modality</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/rheonomy+modality">rheonomy modality</a></p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⇉</mo><mo>⊣</mo><mo>⇝</mo><mo>⊣</mo><mi>Rh</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)</annotation></semantics></math></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/orbifold+cohomology">singular cohesion</a></strong></p> <div id="Diagram" class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>id</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>id</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>fermionic</mi></mover></mtd> <mtd><mo>⇉</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>bosonic</mi></mover></mtd> <mtd><mo>⇝</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi mathvariant="normal">R</mi><mspace width="negativethinmathspace"></mspace><mspace width="negativethinmathspace"></mspace><mi mathvariant="normal">h</mi></mtd> <mtd><mover><mrow></mrow><mi>rheonomic</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>reduced</mi></mover></mtd> <mtd><mi>ℜ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>infinitesimal</mi></mover></mtd> <mtd><mi>ℑ</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mi>&</mi></mtd> <mtd><mover><mrow></mrow><mtext>étale</mtext></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>cohesive</mi></mover></mtd> <mtd><mo lspace="0em" rspace="thinmathspace">ʃ</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♭</mo></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd> <mtd></mtd> <mtd><mo>⊥</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mover><mrow></mrow><mi>discrete</mi></mover></mtd> <mtd><mo>♭</mo></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>♯</mo></mtd> <mtd><mover><mrow></mrow><mi>continuous</mi></mover></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd> <mtd></mtd> <mtd><mo>∨</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>∅</mi></mtd> <mtd><mo>⊣</mo></mtd> <mtd><mo>*</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{&#233;tale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast } </annotation></semantics></math></div></div> <p id="models_2"><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Models+for+Smooth+Infinitesimal+Analysis">Models for Smooth Infinitesimal Analysis</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup></mrow><annotation encoding="application/x-tex">C^\infty</annotation></semantics></math>-ring)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+locus">smooth locus</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fermat+theory">Fermat theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/formal+smooth+%E2%88%9E-groupoid">formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a>, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth ∞-group</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/differential+equations">differential equations</a>, <a class="existingWikiWord" href="/nlab/show/variational+calculus">variational calculus</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-geometry">D-geometry</a>, <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/jet+bundle">jet bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/variational+bicomplex">variational bicomplex</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+complex">Euler-Lagrange complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equation">Euler-Lagrange equation</a>, <a class="existingWikiWord" href="/nlab/show/de+Donder-Weyl+formalism">de Donder-Weyl formalism</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Chern-Weil+theory">Chern-Weil theory</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>, <a class="existingWikiWord" href="/nlab/show/connection+on+an+%E2%88%9E-bundle">connection on an ∞-bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>, <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+K-theory">differential K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+cobordism+cohomology">differential cobordism cohomology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>, <a class="existingWikiWord" href="/nlab/show/fiber+integration+in+differential+cohomology">fiber integration in differential cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomy">holonomy</a>, <a class="existingWikiWord" href="/nlab/show/higher+holonomy">higher holonomy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a>, <a class="existingWikiWord" href="/nlab/show/higher+gauge+theory">higher gauge theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Wilson+line">Wilson line</a>, <a class="existingWikiWord" href="/nlab/show/Wilson+surface">Wilson surface</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a> (<a class="existingWikiWord" href="/nlab/show/super+Cartan+geometry">super</a>, <a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher</a>)</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Klein+geometry">Klein geometry</a>, (<a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+geometry">Euclidean geometry</a>, <a class="existingWikiWord" href="/nlab/show/hyperbolic+geometry">hyperbolic geometry</a>, <a class="existingWikiWord" href="/nlab/show/elliptic+geometry">elliptic geometry</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/pseudo-Riemannian+geometry">pseudo</a>-)<a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+structure">orthogonal structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/isometry">isometry</a>, <a class="existingWikiWord" href="/nlab/show/Killing+vector+field">Killing vector field</a>, <a class="existingWikiWord" href="/nlab/show/Killing+spinor">Killing spinor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a>, <a class="existingWikiWord" href="/nlab/show/super-spacetime">super-spacetime</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+geometry">conformal geometry</a></p> </li> </ul> </div></div> <h4 id="manifolds_and_cobordisms">Manifolds and cobordisms</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a></strong> and <strong><a class="existingWikiWord" href="/nlab/show/cobordisms">cobordisms</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/cobordism+theory">cobordism theory</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Cobordism+and+Complex+Oriented+Cohomology">Introduction</a></em></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+Euclidean+space">locally Euclidean space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coordinate+chart">coordinate chart</a>, <a class="existingWikiWord" href="/nlab/show/coordinate+transformation">coordinate transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/atlas">atlas</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifold">manifold</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+manifold">topological manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+manifold">differentiable manifold</a>, ,<a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinite+dimensional+manifold">infinite dimensional manifold</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Banach+manifold">Banach manifold</a>, <a class="existingWikiWord" href="/nlab/show/Hilbert+manifold">Hilbert manifold</a>, <a class="existingWikiWord" href="/nlab/show/ILH+manifold">ILH manifold</a>, <a class="existingWikiWord" href="/nlab/show/Frechet+manifold">Frechet manifold</a>, <a class="existingWikiWord" href="/nlab/show/convenient+manifold">convenient manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+bundle">tangent bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/normal+bundle">normal bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>, <a class="existingWikiWord" href="/nlab/show/torsion+of+a+G-structure">torsion of a G-structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/orientation">orientation</a>, <a class="existingWikiWord" href="/nlab/show/spin+structure">spin structure</a>, <a class="existingWikiWord" href="/nlab/show/string+structure">string structure</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+structure">fivebrane structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartan+geometry">Cartan geometry</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/B-bordism">B-bordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+cobordism">extended cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+category">cobordism category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/FQFT">functorial quantum field theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom+spectrum">Thom spectrum</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+ring">cobordism ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/genus">genus</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Genera and invariants</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/signature+genus">signature genus</a>, <a class="existingWikiWord" href="/nlab/show/Kervaire+invariant">Kervaire invariant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-hat+genus">A-hat genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> <p><strong>Classification</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-manifolds">2-manifolds</a>/<a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/3-manifolds">3-manifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kirby+calculus">Kirby calculus</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/4-manifolds">4-manifolds</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dehn+surgery">Dehn surgery</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exotic+smooth+structure">exotic smooth structure</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitney+embedding+theorem">Whitney embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thom%27s+transversality+theorem">Thom's transversality theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pontrjagin-Thom+construction">Pontrjagin-Thom construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galatius-Tillmann-Madsen-Weiss+theorem">Galatius-Tillmann-Madsen-Weiss theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometrization+conjecture">geometrization conjecture</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poincar%C3%A9+conjecture">Poincaré conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptization+conjecture">elliptization conjecture</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#basic_facts'>Basic facts</a></li> <li><a href='#complexified_differentials'>Complexified differentials</a></li> <li><a href='#PicardGroupOfHolomorphicLineBundles'>Picard group of holomorphic line bundles</a></li> <li><a href='#central_theorems'>Central theorems</a></li> <li><a href='#homotopy_type'>Homotopy type</a></li> <li><a href='#branched_covers'>Branched covers</a></li> <li><a href='#function_field_analogy'>Function field analogy</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A Riemann surface is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebro-geometric</a> object with good properties. The name ‘surface’ comes from the classical case, which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-dimensional over the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a> and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-dimensional over the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>.</p> <p>There are several distinct meaning of what is a Riemann surface, and it can be considered in several generalities. But the main definition by far is the classical one.</p> <h2 id="definition">Definition</h2> <p>Classically, a <strong>Riemann surface</strong> is a <a class="existingWikiWord" href="/nlab/show/connected+space">connected</a> complex-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, in the strictest sense of ‘manifold’. In other words, it’s a <a class="existingWikiWord" href="/nlab/show/Hausdorff+space">Hausdorff</a> <a class="existingWikiWord" href="/nlab/show/second+countable+space">second countable space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> which is locally <a class="existingWikiWord" href="/nlab/show/homeomorphism">homeomorphic</a> to the <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> via charts (i.e., <a class="existingWikiWord" href="/nlab/show/homeomorphisms">homeomorphisms</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\phi_i:U_i \to V_i</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>i</mi></msub><mo>⊂</mo><mi>M</mi><mo>,</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>⊂</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">U_i \subset M, V_i \subset \mathbb{C}</annotation></semantics></math> open and such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>j</mi></msub><mo>∘</mo><msubsup><mi>ϕ</mi> <mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>:</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>j</mi></msub><mo>→</mo><msub><mi>V</mi> <mi>i</mi></msub><mo>∩</mo><msub><mi>V</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">\phi_j \circ \phi_i^{-1}: V_i \cap V_j \to V_i \cap V_j</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/holomorphic+function">holomorphic</a>.</p> <p>It is not necessary to make any assumption about whether there exists a countable base for the topology (second countable) or whether it has a countable dense subset (separable). This is because Tibor Rado (1923) proved that all Riemann surfaces, without such prior assumptions, must necessarily have a countable base. Thus for elegance this condition is customarily <em>not</em> assumed by specialists in Riemann surfaces.</p> <p>There are generalizations, e.g., over <a class="existingWikiWord" href="/nlab/show/local+field">local field</a>s in <a class="existingWikiWord" href="/nlab/show/rigid+analytic+geometry">rigid analytic geometry</a>.</p> <h2 id="examples">Examples</h2> <p>Evidently an <a class="existingWikiWord" href="/nlab/show/open+subspace">open subspace</a> of a Riemann surface is a Riemann surface. In particular, an open subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math> is a Riemann surface in a natural manner.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>P</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>ℂ</mi><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">P^1(\mathbb{C}) := \mathbb{C} \cup \{ \infty \}</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">S^2</annotation></semantics></math> is a Riemann surface with the open sets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>=</mo><mi>ℂ</mi><mo>,</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>=</mo><mi>ℂ</mi><mo>−</mo><mo stretchy="false">{</mo><mn>0</mn><mo stretchy="false">}</mo><mo>∪</mo><mo stretchy="false">{</mo><mn>∞</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U_1 = \mathbb{C}, U_2 = \mathbb{C} - \{0\} \cup \{\infty\}</annotation></semantics></math> and the charts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>=</mo><mi>z</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>=</mo><mfrac><mn>1</mn><mi>z</mi></mfrac><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi_1 =z, \;\phi_2 = \frac{1}{z}.</annotation></semantics></math></div> <p>The transition map is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mn>1</mn><mi>z</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{1}{z}</annotation></semantics></math> and thus holomorphic on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mn>1</mn></msub><mo>∩</mo><msub><mi>U</mi> <mn>2</mn></msub><mo>=</mo><msup><mi>ℂ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">U_1 \cap U_2 = \mathbb{C}^*</annotation></semantics></math>.</p> <p>An important example comes from <a class="existingWikiWord" href="/nlab/show/analytic+continuation">analytic continuation</a>, which we will briefly sketch below. A <strong>function element</strong> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,V)</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">f: V \to \mathbb{C}</annotation></semantics></math> is holomorphic and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊂</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">V \subset \mathbb{C}</annotation></semantics></math> is an open disk. Two function elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,V), (g,W)</annotation></semantics></math> are said to be <strong>direct analytic continuations</strong> of each other if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∩</mo><mi>W</mi><mo>≠</mo><mi>∅</mi></mrow><annotation encoding="application/x-tex">V \cap W \neq \emptyset</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>≡</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \equiv g </annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∩</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">V \cap W</annotation></semantics></math>. By piecing together direct analytic continuations on a curve, we can talk about the analytic continuation of a function element along a curve (which may or may not exist, but if it does, it is unique).</p> <p>Starting with a given function element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\gamma = (f,V)</annotation></semantics></math>, we can consider the totality <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> of all equivalence classes of function elements that can be obtained by continuing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> along curves in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is actually a Riemann surface.</p> <p>Indeed, we must first put a topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">(g,W) \in X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>=</mo><msub><mi>D</mi> <mi>r</mi></msub><mo stretchy="false">(</mo><msub><mi>w</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W=D_r(w_0)</annotation></semantics></math> centered at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>w</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">w_0</annotation></semantics></math>, then let a neighborhood of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> be given by all function elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>g</mi> <mi>w</mi></msub><mo>,</mo><mi>W</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(g_w, W')</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>∈</mo><mi>W</mi><mo>,</mo><mi>W</mi><mo>′</mo><mo>⊂</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">w \in W, W' \subset W</annotation></semantics></math>; these form a basis for a suitable topology on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Then the coordinate projections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>w</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">(g,W) \to w_0</annotation></semantics></math> form appropriate local coordinates. In fact, there is a globally defined map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">X \to \mathbb{C}</annotation></semantics></math>, whose image in general will be a proper subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">\mathbb{C}</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="basic_facts">Basic facts</h3> <p>Since we have local coordinates, we can define a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f: X \to Y</annotation></semantics></math> of Riemann surfaces to be <strong>holomorphic</strong> or <strong>regular</strong> if it is so in local coordinates. In particular, we can define a holomorphic complex function as a holomorphic map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">f: X \to \mathbb{C}</annotation></semantics></math>; for <a class="existingWikiWord" href="/nlab/show/meromorphic+function">meromorphicity</a>, this becomes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><msup><mi>S</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">f: X \to S^2</annotation></semantics></math>.</p> <p>Many of the usual theorems of elementary <a class="existingWikiWord" href="/nlab/show/complex+analysis">complex analysis</a> (that is to say, the local ones) transfer immediately to the case of Riemann surfaces. For instance, we can locally get a <a class="existingWikiWord" href="/nlab/show/Laurent+series">Laurent expansion</a>, etc.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f: X \to Y</annotation></semantics></math> be a regular map. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is compact and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is nonconstant, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is surjective and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> compact.</p> </div> <p>To see this, note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(X)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a>, and an open subset by the <span class="newWikiWord">open mapping theorem<a href="/nlab/new/open+mapping+theorem">?</a></span>, so the result follows by connectedness of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> <h3 id="complexified_differentials">Complexified differentials</h3> <p>Since a Riemann surface <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> in the usual (real) sense, it is possible to do the usual <a class="existingWikiWord" href="/nlab/show/differential+form">exterior calculus</a>. We could consider a 1-form to be a section of the (usual) <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T^*(X)</annotation></semantics></math>, but it is more natural to take the <strong>complexified cotangent bundle</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msup><mi>T</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C} \otimes_{\mathbb{R}} T^*(X)</annotation></semantics></math>, which we will in the future just abbreviate <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T^*(X)</annotation></semantics></math>; this should not be confusing since we will only do this when we talk about complex manifolds. Sections of this bundle will be called (complex-valued) 1-forms. Similarly, we do the same for 2-forms.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>=</mo><mi>x</mi><mo>+</mo><mi>i</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">z = x + i y</annotation></semantics></math> is a local coordinate on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, defined say on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">U \subset X</annotation></semantics></math>, define the (complex) differentials</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>z</mi><mo>=</mo><mi>d</mi><mi>x</mi><mo>+</mo><mi>i</mi><mi>d</mi><mi>y</mi><mo>,</mo><mspace width="thickmathspace"></mspace><mi>d</mi><mover><mi>z</mi><mo stretchy="false">¯</mo></mover><mo>=</mo><mi>d</mi><mi>x</mi><mo>−</mo><mi>i</mi><mi>d</mi><mi>y</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">d z = d x + i d y , \;d\bar{z} = d x - i d y.</annotation></semantics></math></div> <p>These form a basis for the complexified cotangent space at each point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. There is also a dual basis</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mo>:</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>−</mo><mi>i</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mspace width="thickmathspace"></mspace><mfrac><mo>∂</mo><mrow><mo>∂</mo><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow></mfrac><mo>:</mo><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mrow><mo>(</mo><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>x</mi></mrow></mfrac><mo>+</mo><mi>i</mi><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>y</mi></mrow></mfrac><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \frac{\partial}{\partial z } := \frac{1}{2}\left( \frac{\partial}{\partial x} - i \frac{\partial}{\partial y}\right), \; \frac{\partial}{\partial \bar{z} } := \frac{1}{2}\left( \frac{\partial}{\partial x} + i \frac{\partial}{\partial y}\right) </annotation></semantics></math></div> <p>for the complexified tangent space.</p> <p>We now claim that we can split the tangent space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>T</mi> <mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mi>T</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T(X) = T^{1,0}(X) + T^{0,1}(X)</annotation></semantics></math>, where the former consists of multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial}{\partial z}</annotation></semantics></math> and the latter of multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial}{\partial \bar{z}}</annotation></semantics></math>; clearly a similar thing is possible for the cotangent space. This is always possible locally, and a holomorphic map preserves the decomposition. One way to see the last claim quickly is that given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>:</mo><mi>U</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">g: U \to \mathbb{C}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>⊂</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">U \subset \mathbb{C}</annotation></semantics></math> open and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">0 \in U</annotation></semantics></math> (just for convenience), we can write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mi>Az</mi><mo>+</mo><mi>A</mi><mo>′</mo><mover><mi>z</mi><mo stretchy="false">¯</mo></mover><mo>+</mo><mi>o</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> g(z) = g(0) + Az + A' \bar{z} + o(|z|) </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>g</mi></mrow><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>A</mi><mo>′</mo><mo>=</mo><mfrac><mrow><mo>∂</mo><mi>g</mi></mrow><mrow><mo>∂</mo><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow></mfrac><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A = \frac{\partial g }{\partial z }(0), A' = \frac{\partial g }{\partial \bar{z} }(0)</annotation></semantics></math>, which we will often abbreviate as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>z</mi></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mi>g</mi> <mover><mi>z</mi><mo stretchy="false">¯</mo></mover></msub><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_z(0), g_{\bar{z}}(0)</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ψ</mi><mo>:</mo><mi>U</mi><mo>′</mo><mo>→</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">\psi: U' \to U</annotation></semantics></math> is holomorphic and conformal sending <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>z</mi> <mn>0</mn></msub><mo>∈</mo><mi>U</mi><mo>′</mo><mo>→</mo><mn>0</mn><mo>∈</mo><mi>U</mi></mrow><annotation encoding="application/x-tex">z_0 \in U' \to 0 \in U</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>ζ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>+</mo><mi>A</mi><mi>ϕ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ζ</mi><mo>−</mo><msub><mi>z</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>A</mi><mo>′</mo><mover><mrow><mi>ϕ</mi><mo>′</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ζ</mi><mo>−</mo><msub><mi>z</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><mo>¯</mo></mover><mo>+</mo><mi>o</mi><mo stretchy="false">(</mo><mo stretchy="false">|</mo><mi>z</mi><mo stretchy="false">|</mo><mo stretchy="false">)</mo><mo>;</mo></mrow><annotation encoding="application/x-tex"> g(\phi(\zeta)) = g(\phi(0)) + A \phi'(z_0)(\zeta-z_0) + A' \overline{ \phi'(z_0)(\zeta-z_0)} + o(|z|); </annotation></semantics></math></div> <p>in particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> preserves the decomposition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>T</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T_0(\mathbb{C})</annotation></semantics></math>.</p> <p>Given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">f: X \to \mathbb{C}</annotation></semantics></math> smooth, we can consider the projections of the 1-form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>df</mi></mrow><annotation encoding="application/x-tex">df</annotation></semantics></math> onto <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T^{1,0}(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T^{0,1}(X)</annotation></semantics></math>, respectively; these will be called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>f</mi><mo>,</mo><mover><mo>∂</mo><mo>¯</mo></mover><mi>f</mi></mrow><annotation encoding="application/x-tex">\partial f, \overline{\partial} f</annotation></semantics></math>. Similarly, we define the corresponding operators on 1-forms: to define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>ω</mi></mrow><annotation encoding="application/x-tex">\partial \omega</annotation></semantics></math>, first project onto <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T^{0,1}(M)</annotation></semantics></math> (the reversal is intentional!) and then apply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi></mrow><annotation encoding="application/x-tex">d</annotation></semantics></math>, and vice versa for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∂</mo><mo>¯</mo></mover><mi>ω</mi></mrow><annotation encoding="application/x-tex">\overline{\partial} \omega</annotation></semantics></math>.</p> <p>In particular, if we write in local coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>=</mo><mi>u</mi><mi>d</mi><mi>z</mi><mo>+</mo><mi>v</mi><mi>d</mi><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">\omega = u d z + v d\bar{z}</annotation></semantics></math>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∂</mo><mi>ω</mi><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>v</mi><mi>d</mi><mover><mi>z</mi><mo stretchy="false">¯</mo></mover><mo stretchy="false">)</mo><mo>=</mo><msub><mi>v</mi> <mi>z</mi></msub><mi>d</mi><mi>z</mi><mo>∧</mo><mi>d</mi><mover><mi>z</mi><mo stretchy="false">¯</mo></mover><mo>,</mo></mrow><annotation encoding="application/x-tex"> \partial \omega = d( v d \bar{z}) = v_z d z \wedge d\bar{z},</annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mo>∂</mo><mo>¯</mo></mover><mi>ω</mi><mo>=</mo><mi>d</mi><mo stretchy="false">(</mo><mi>u</mi><mi>d</mi><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>u</mi> <mover><mi>z</mi><mo stretchy="false">¯</mo></mover></msub><mi>d</mi><mover><mi>z</mi><mo stretchy="false">¯</mo></mover><mo>∧</mo><mi>d</mi><mi>z</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> \overline{\partial} \omega = d( u d z) = u_{\bar{z}} d\bar{z} \wedge d z.</annotation></semantics></math></div> <p>To see this, we have tacitly observed that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mi>v</mi><mo>=</mo><msub><mi>v</mi> <mi>z</mi></msub><mi>d</mi><mi>z</mi><mo>+</mo><msub><mi>v</mi> <mover><mi>z</mi><mo stretchy="false">¯</mo></mover></msub><mi>d</mi><mover><mi>z</mi><mo stretchy="false">¯</mo></mover></mrow><annotation encoding="application/x-tex">d v = v_z d z + v_{\bar{z}} d\bar{z}</annotation></semantics></math>.</p> <h3 id="PicardGroupOfHolomorphicLineBundles">Picard group of holomorphic line bundles</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Picard+group">Picard group</a> of a Riemann surface is the group of <a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundles">holomorphic line bundles</a> in it. Introductions include (<a href="#Bobenko">Bobenko, section 8</a>).</p> <p>See also at <em><a class="existingWikiWord" href="/nlab/show/Narasimhan%E2%80%93Seshadri+theorem">Narasimhan–Seshadri theorem</a></em> and at <em><a href="moduli+space+of+connections#FlatConnectionsOverATorus">moduli space of connections – Flat connections over a torus</a></em>.</p> <h3 id="central_theorems">Central theorems</h3> <p>In the theory of Riemann surfaces, there are several important theorems. Here are two:</p> <ul> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Riemann-Roch+theorem">Riemann-Roch theorem</a>, which analyzes the vector space of meromorphic functions satisfying certain conditions on zeros and poles;</p> </li> <li> <p>The <span class="newWikiWord">uniformization theorem<a href="/nlab/new/uniformization+theorem">?</a></span>, which partially classifies Riemann surfaces.</p> </li> </ul> <h3 id="homotopy_type">Homotopy type</h3> <p>A <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> Riemann surface of <a class="existingWikiWord" href="/nlab/show/genus">genus</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">g \geq 2</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a>. The <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> is a <a class="existingWikiWord" href="/nlab/show/Fuchsian+group">Fuchsian group</a>.</p> <p>(<a href="http://mathoverflow.net/a/93340/381">MO discussion</a>)</p> <h3 id="branched_covers">Branched covers</h3> <p>By the <a class="existingWikiWord" href="/nlab/show/Riemann+existence+theorem">Riemann existence theorem</a>, every connected compact <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a> admits the <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of a <a class="existingWikiWord" href="/nlab/show/branched+cover+of+the+Riemann+sphere">branched cover of the Riemann sphere</a>. (<a href="http://mathoverflow.net/a/53286/381">MO discussion</a>)</p> <h3 id="function_field_analogy">Function field analogy</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/function+field+analogy">function field analogy</a></strong></p> <table><thead><tr><th></th><th><a class="existingWikiWord" href="/nlab/show/number+fields">number fields</a> (“<a class="existingWikiWord" href="/nlab/show/function+fields">function fields</a> of <a class="existingWikiWord" href="/nlab/show/curves">curves</a> over <a class="existingWikiWord" href="/nlab/show/F1">F1</a>”)</th><th><a class="existingWikiWord" href="/nlab/show/function+fields">function fields</a> of <a class="existingWikiWord" href="/nlab/show/curves">curves</a> over <a class="existingWikiWord" href="/nlab/show/finite+fields">finite fields</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_q</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/arithmetic+curves">arithmetic curves</a>)</th><th><a class="existingWikiWord" href="/nlab/show/Riemann+surfaces">Riemann surfaces</a>/<a class="existingWikiWord" href="/nlab/show/complex+curves">complex curves</a></th></tr></thead><tbody><tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/affine+line">affine</a> and <a class="existingWikiWord" href="/nlab/show/projective+line">projective line</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/integers">integers</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q[z]</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/polynomials">polynomials</a>, <a class="existingWikiWord" href="/nlab/show/polynomial+algebra">polynomial algebra</a> on <a class="existingWikiWord" href="/nlab/show/affine+line">affine line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1_{\mathbb{F}_q}</annotation></semantics></math>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{\mathbb{C}}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q(z)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/rational+fractions">rational fractions</a>/<a class="existingWikiWord" href="/nlab/show/rational+function+on+an+affine+variety">rational function on affine line</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1_{\mathbb{F}_q}</annotation></semantics></math>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/meromorphic+functions">meromorphic functions</a> on <a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/prime+number">prime number</a>/non-archimedean <a class="existingWikiWord" href="/nlab/show/place">place</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">x \in \mathbb{F}_p</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>−</mo><mi>x</mi><mo>∈</mo><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">z - x \in \mathbb{F}_q[z]</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/irreducible+element">irreducible</a> <a class="existingWikiWord" href="/nlab/show/monic+polynomial">monic polynomial</a> of <a class="existingWikiWord" href="/nlab/show/degree+of+a+polynomial">degree</a> one</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>ℂ</mi></mrow><annotation encoding="application/x-tex">x \in \mathbb{C}</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>−</mo><mi>x</mi><mo>∈</mo><msub><mi>𝒪</mi> <mi>ℂ</mi></msub></mrow><annotation encoding="application/x-tex">z - x \in \mathcal{O}_{\mathbb{C}}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/function">function</a> which <a class="existingWikiWord" href="/nlab/show/subtracts">subtracts</a> the <a class="existingWikiWord" href="/nlab/show/complex+number">complex number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> from the <a class="existingWikiWord" href="/nlab/show/variable">variable</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi></mrow><annotation encoding="application/x-tex">z</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/place+at+infinity">place at infinity</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(\mathbb{Z})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Spec%28Z%29">Spec(Z)</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>𝔸</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow> <mn>1</mn></msubsup></mrow><annotation encoding="application/x-tex">\mathbb{A}^1_{\mathbb{F}_q}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/affine+line">affine line</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/complex+plane">complex plane</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo><mo>∪</mo><msub><mi>place</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">Spec(\mathbb{Z}) \cup place_{\infty}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℙ</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{P}_{\mathbb{F}_q}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/projective+line">projective line</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemann+sphere">Riemann sphere</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∂</mo> <mi>p</mi></msub><mo>≔</mo><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>−</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mi>p</mi></mfrac></mrow><annotation encoding="application/x-tex">\partial_p \coloneqq \frac{(-)^p - (-)}{p}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Fermat+quotient">Fermat quotient</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial}{\partial z}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/coordinate">coordinate</a> <a class="existingWikiWord" href="/nlab/show/derivation">derivation</a>)</td><td style="text-align: left;">“</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+a+number+field">genus of the rational numbers</a> = 0</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of the Riemann sphere</a> = 0</td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/formal+neighbourhoods">formal neighbourhoods</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msup><mi>p</mi> <mi>n</mi></msup><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}/(p^n \mathbb{Z})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/prime+power+local+ring">prime power local ring</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q [z]/((z-x)^n \mathbb{F}_q [z])</annotation></semantics></math> (<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>-th order univariate <a class="existingWikiWord" href="/nlab/show/local+Artinian+ring">local Artinian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>𝔽</mi> <mi>q</mi></msub> </mrow> <annotation encoding="application/x-tex">\mathbb{F}_q</annotation> </semantics> </math>-algebra</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mi>ℂ</mi><mo stretchy="false">[</mo><mi>z</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}[z]/((z-x)^n \mathbb{C}[z])</annotation></semantics></math> (<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>-th order univariate <a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>ℂ</mi> </mrow> <annotation encoding="application/x-tex">\mathbb{C}</annotation> </semantics> </math>-algebra</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_p</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/p-adic+integers">p-adic integers</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q[ [ z -x ] ]</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/power+series">power series</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}[ [z-x] ]</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spf</mi><mo stretchy="false">(</mo><msub><mi>ℤ</mi> <mi>p</mi></msub><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow></munder><mi>X</mi></mrow><annotation encoding="application/x-tex">Spf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X</annotation></semantics></math> (“<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/arithmetic+jet+space">arithmetic jet space</a>” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/formal+disks">formal disks</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℚ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{Q}_p</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/p-adic+numbers">p-adic numbers</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{F}_q((z-x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/Laurent+series">Laurent series</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}((z-x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/holomorphic+functions">holomorphic functions</a> on punctured <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔸</mi> <mi>ℚ</mi></msub><mo>=</mo><munder><mrow><msup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><mrow><mi>p</mi><mspace width="thickmathspace"></mspace><mi>place</mi></mrow></munder><msub><mi>ℚ</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/ring+of+adeles">ring of adeles</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔸</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{A}_{\mathbb{F}_q((t))}</annotation></semantics></math> ( <a href="ring%20of%20adeles#ForAGlobalField">adeles of function field</a> )</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mrow><msup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><mrow><mi>x</mi><mo>∈</mo><mi>ℂ</mi></mrow></munder><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((z-x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/restricted+product">restricted product</a> of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕀</mi> <mi>ℚ</mi></msub><mo>=</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>ℚ</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/group+of+ideles">group of ideles</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕀</mi> <mrow><msub><mi>𝔽</mi> <mi>q</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{I}_{\mathbb{F}_q((t))}</annotation></semantics></math> ( <a class="existingWikiWord" href="/nlab/show/group+of+ideles">ideles of function field</a> )</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mrow><msup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup></mrow><mrow><mi>x</mi><mo>∈</mo><mi>ℂ</mi></mrow></munder><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>z</mi><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((z-x)))</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/theta+functions">theta functions</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Jacobi+theta+function">Jacobi theta function</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/zeta+functions">zeta functions</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Riemann+zeta+function">Riemann zeta function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Goss+zeta+function">Goss zeta function</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/branched+covering">branched covering</a> curves</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/number+field">number field</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mo>↪</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q} \hookrightarrow K</annotation></semantics></math> a possibly <a class="existingWikiWord" href="/nlab/show/ramified">ramified</a> <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> <a class="existingWikiWord" href="/nlab/show/field+extension">field extension</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/function+field">function field</a> of an <a class="existingWikiWord" href="/nlab/show/algebraic+curve">algebraic curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔽</mi> <mi>p</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{F}_p</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">K_\Sigma</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/sheaf+of+rational+functions">sheaf of rational functions</a> on <a class="existingWikiWord" href="/nlab/show/complex+curve">complex curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_K</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>Σ</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{\Sigma}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/structure+sheaf">structure sheaf</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Spec</mi> <mi>an</mi></msub><mo stretchy="false">(</mo><msub><mi>𝒪</mi> <mi>K</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/spectrum+of+a+commutative+ring">spectrum</a> with archimedean <a class="existingWikiWord" href="/nlab/show/places">places</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/arithmetic+curve">arithmetic curve</a>)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi><mo>→</mo><mi>ℂ</mi><msup><mi>P</mi> <mn>1</mn></msup></mrow><annotation encoding="application/x-tex">\Sigma \to \mathbb{C}P^1</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/complex+curve">complex curve</a> being <a class="existingWikiWord" href="/nlab/show/branched+cover+of+Riemann+sphere">branched cover of Riemann sphere</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>p</mi></msup><mo>−</mo><mi>Φ</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><mi>p</mi></mfrac></mrow><annotation encoding="application/x-tex">\frac{(-)^p - \Phi(-)}{p}</annotation></semantics></math> (lift of <a class="existingWikiWord" href="/nlab/show/Frobenius+morphism">Frobenius morphism</a>/<a class="existingWikiWord" href="/nlab/show/Lambda-ring">Lambda-ring</a> structure)</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mfrac><mo>∂</mo><mrow><mo>∂</mo><mi>z</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac{\partial}{\partial z}</annotation></semantics></math></td><td style="text-align: left;">“</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+a+number+field">genus of a number field</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+an+algebraic+curve">genus of an algebraic curve</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/genus+of+a+surface">genus of a surface</a></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/formal+neighbourhoods">formal neighbourhoods</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> prime ideal in <a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_K</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">x \in \Sigma</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow><annotation encoding="application/x-tex">x \in \Sigma</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mi>v</mi></msub></mrow><annotation encoding="application/x-tex">K_v</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a> at <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}((z_x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a> on punctured <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒪</mi> <mrow><msub><mi>K</mi> <mi>v</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\mathcal{O}_{K_v}</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/ring+of+integers">ring of integers</a> of <a class="existingWikiWord" href="/nlab/show/formal+completion">formal completion</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}[ [ z_x ] ]</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a> on <a class="existingWikiWord" href="/nlab/show/formal+disk">formal disk</a> around <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔸</mi> <mi>K</mi></msub></mrow><annotation encoding="application/x-tex">\mathbb{A}_K</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/ring+of+adeles">ring of adeles</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod^\prime_{x\in \Sigma} \mathbb{C}((z_x))</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/restricted+product">restricted product</a> of <a class="existingWikiWord" href="/nlab/show/function+rings">function rings</a> on all punctured <a class="existingWikiWord" href="/nlab/show/formal+disks">formal disks</a> around all points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow></msub><mi>ℂ</mi><mo stretchy="false">[</mo><mo stretchy="false">[</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">]</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\prod_{x\in \Sigma} \mathbb{C}[ [z_x] ] </annotation></semantics></math> (function ring on all <a class="existingWikiWord" href="/nlab/show/formal+disks">formal disks</a> around all points in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝕀</mi> <mi>K</mi></msub><mo>=</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>K</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{I}_K = GL_1(\mathbb{A}_K)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/group+of+ideles">group of ideles</a>)</td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>x</mi><mo>∈</mo><mi>Σ</mi></mrow> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>z</mi> <mi>x</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((z_x)))</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Galois+group">Galois group</a></td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_1(\Sigma)</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Galois+representation">Galois representation</a></td><td style="text-align: left;">“</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flat+connection">flat connection</a> (“<a class="existingWikiWord" href="/nlab/show/local+system">local system</a>”) on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/class+field+theory">class field theory</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/class+field+theory">class field theory</a></td><td style="text-align: left;">“</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+class+field+theory">geometric class field theory</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hilbert+reciprocity+law">Hilbert reciprocity law</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Artin+reciprocity+law">Artin reciprocity law</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Weil+reciprocity+law">Weil reciprocity law</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>\</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>K</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(K)\backslash GL_1(\mathbb{A}_K)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/idele+class+group">idele class group</a>)</td><td style="text-align: left;">“</td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>\</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>K</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><msub><mi>GL</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathcal{O})</annotation></semantics></math></td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bun</mi> <mrow><msub><mi>GL</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bun_{GL_1}(\Sigma)</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/moduli+stack+of+bundles">moduli stack of line bundles</a>, by <a class="existingWikiWord" href="/nlab/show/Weil+uniformization+theorem">Weil uniformization theorem</a>)</td></tr> <tr><td style="text-align: left;"><strong>non-abelian class field theory and automorphy</strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;">number field <a class="existingWikiWord" href="/nlab/show/Langlands+correspondence">Langlands correspondence</a></td><td style="text-align: left;">function field <a class="existingWikiWord" href="/nlab/show/Langlands+correspondence">Langlands correspondence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/geometric+Langlands+correspondence">geometric Langlands correspondence</a></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>\</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>𝔸</mi> <mi>K</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">/</mo><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O})</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/constant+sheaves">constant sheaves</a> on this <a class="existingWikiWord" href="/nlab/show/stack">stack</a> form <a class="existingWikiWord" href="/nlab/show/unramified">unramified</a> <a class="existingWikiWord" href="/nlab/show/automorphic+representations">automorphic representations</a>)</td><td style="text-align: left;">“</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Bun</mi> <mrow><msub><mi>GL</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>ℂ</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>Σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Bun_{GL_n(\mathbb{C})}(\Sigma)</annotation></semantics></math> (<a href="moduli+space+of+bundles#OverCurvesAndTheLanglandsCorrespondence">moduli stack of bundles on the curve</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math>, by <a class="existingWikiWord" href="/nlab/show/Weil+uniformization+theorem">Weil uniformization theorem</a>)</td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a href="Weil+conjecture+on+Tamagawa+numbers#NumberFieldCase">Tamagawa-Weil for number fields</a></td><td style="text-align: left;"><a href="Weil+conjecture+on+Tamagawa+numbers#FunctionFieldCase">Tamagawa-Weil for function fields</a></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/theta+functions">theta functions</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hecke+theta+function">Hecke theta function</a></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/functional+determinant">functional determinant</a> <a class="existingWikiWord" href="/nlab/show/determinant+line+bundle">line bundle</a> of <a class="existingWikiWord" href="/nlab/show/Dirac+operator">Dirac operator</a>/chiral <a class="existingWikiWord" href="/nlab/show/Laplace+operator">Laplace operator</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/zeta+functions">zeta functions</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Dedekind+zeta+function">Dedekind zeta function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Weil+zeta+function">Weil zeta function</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/zeta+function+of+a+Riemann+surface">zeta function of a Riemann surface</a>/<a class="existingWikiWord" href="/nlab/show/zeta+function+of+an+elliptic+differential+operator">of the Laplace operator</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Σ</mi></mrow><annotation encoding="application/x-tex">\Sigma</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/higher+dimensional+arithmetic+geometry">higher dimensional spaces</a></strong></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><strong><a class="existingWikiWord" href="/nlab/show/zeta+functions">zeta functions</a></strong></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hasse-Weil+zeta+function">Hasse-Weil zeta function</a></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/annulus">annulus</a>, <a class="existingWikiWord" href="/nlab/show/2-sphere">2-sphere</a>, <a class="existingWikiWord" href="/nlab/show/trinion">trinion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Teichm%C3%BCller+space">Teichmüller space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+Riemann+surfaces">moduli space of Riemann surfaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+vector+bundle">stable vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/worldsheet">worldsheet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/beta-gamma+system">beta-gamma system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quadratic+differential">quadratic differential</a></p> </li> </ul> <h2 id="references">References</h2> <p>Historical references:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Hermann+Weyl">Hermann Weyl</a>, <em>Die Idee der Riemannschen Fläche</em>, 1913 (<em>The concept of a Riemann surface</em>) (on the book, by Peter Schreiber, 2013: <a href="http://mathineurope.eu/en/home/47-information/math-calendar/971-1913-publication-of-the-concept-of-a-riemann-surface-by-hermann-weyl">web</a>)</p> </li> <li id="Rado25"> <p><a class="existingWikiWord" href="/nlab/show/Tibor+Rad%C3%B3">Tibor Radó</a>, <em>Über den Begriff der Riemannschen Fläche</em>, Acta Litt. Sci. Szeged, 2 (101-121), 10 (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/rado.pdf">pdf</a>, <a href="http://acta.bibl.u-szeged.hu/13312/1/math_002_101-121.pdf">pdf</a>)</p> <blockquote> <p>(proving the <a class="existingWikiWord" href="/nlab/show/triangulation+theorem">triangulation theorem</a>)</p> </blockquote> </li> </ul> <p>Monograph:</p> <ul> <li id="Gunning66"><a class="existingWikiWord" href="/nlab/show/Robert+C.+Gunning">Robert C. Gunning</a>, <em>Lectures on Riemann Surfaces</em>, Princeton University Press (1966) [<a href="http://ronan.terpereau.perso.math.cnrs.fr/Master_Class_2023_Dijon/GUNNING_Lectures%20on%20Riemann%20Surfaces.pdf">pdf</a>]</li> </ul> <p>Lecture notes:</p> <ul> <li id="Bobenko"> <p><a class="existingWikiWord" href="/nlab/show/Alexander+Bobenko">Alexander Bobenko</a>, <em>Compact Riemann Surfaces</em> lecture notes (<a href="http://page.math.tu-berlin.de/~bobenko/Lehre/Skripte/RS.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eberhard+Freitag">Eberhard Freitag</a>, <em>Riemann surfaces – Sheaf theory, Riemann Surfaces, Automorphic forms</em>, 2013 (<a href="http://www.rzuser.uni-heidelberg.de/~t91/skripten/riemfl.pdf">pdf</a>)</p> </li> <li id="Wehler20"> <p><a class="existingWikiWord" href="/nlab/show/Joachim+Wehler">Joachim Wehler</a>, <em>Riemann surfaces</em> 2020 (<a href="https://www.mathematik.uni-muenchen.de/~wehler/20190530_RiemannSurfacesScript.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 18, 2023 at 05:51:56. 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