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class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <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='#in_terms_of_structure'>In terms 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>-structure</a></li> <li><a href='#InSUBordismTheory'>In SU-bordism theory</a></li> <li><a href='#artinmazur_formal_group'>Artin-Mazur formal group</a></li> <li><a href='#as_supersymmetric_compactification_spaces_in_string_theory'>As supersymmetric compactification spaces in string theory</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>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <em>Calabi-Yau variety</em> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a> with (<a class="existingWikiWord" href="/nlab/show/holomorphic+line+bundle">holomorphically</a>, rather than just topologically) trivial <a class="existingWikiWord" href="/nlab/show/canonical+bundle">canonical bundle</a>. This is equivalent to saying that it is real <a class="existingWikiWord" href="/nlab/show/Riemannian+manifold">Riemannian manifold</a> of even <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>N</mi></mrow><annotation encoding="application/x-tex">2 N</annotation></semantics></math> which has <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a> in the <a class="existingWikiWord" href="/nlab/show/subgroup">subgroup</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>⊂</mo><mi>O</mi><mo stretchy="false">(</mo><mn>2</mn><mi>N</mi><mo>,</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(N)\subset O(2 N, \mathbb{R})</annotation></semantics></math>.</p> <p>For <a class="existingWikiWord" href="/nlab/show/compact+topological+space">compact</a> <a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifolds">Kähler manifolds</a>, <span class="newWikiWord">Yau's theorem<a href="/nlab/new/Yau%27s+theorem">?</a></span> (also known as the <span class="newWikiWord">Calabi conjecture<a href="/nlab/new/Calabi+conjecture">?</a></span>) states any of the above conditions implies the vanishing of the first <a class="existingWikiWord" href="/nlab/show/Chern+class">Chern class</a>.</p> <div class="query"> <p>Is it also true for non-compact?</p> </div> <p>Note that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_1(X) = 0</annotation></semantics></math> implies in general that the canonical bundle is <em>topologically trivial</em>. But 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 a simply connected compact Kähler manifold, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">c_1(X) = 0</annotation></semantics></math> implies further that the canonical bundle is <em>holomorphically trivial</em>.</p> <div class="query"> <p>The language used in this article is implicitly analytic, rather than algebraic. Is this OK? Or should I make this explicit?</p> </div> <h2 id="definition">Definition</h2> <p>A Calabi-Yau variety can be described algebraically as a <a class="existingWikiWord" href="/nlab/show/smooth+scheme">smooth</a> <a class="existingWikiWord" href="/nlab/show/proper+scheme">proper</a> <a class="existingWikiWord" href="/nlab/show/variety">variety</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> of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> <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> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> <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> (not necessarily <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed</a> and not necessarily of <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>) in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>X</mi></msub><mo>=</mo><msup><mo>∧</mo> <mi>n</mi></msup><msup><mi>Ω</mi> <mn>1</mn></msup><mo>≃</mo><msub><mi>𝒪</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">\omega_X=\wedge^n\Omega^1\simeq \mathcal{O}_X</annotation></semantics></math> and also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">H^j(X, \mathcal{O}_X)=0</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>j</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1\leq j \leq n-1</annotation></semantics></math>.</p> <p>If the base field is <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 one can form the analytification 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> and obtain a <a class="existingWikiWord" href="/nlab/show/compact+space">compact</a> <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> that satisfies the first given definition.</p> <p>Beware that there are slightly different (and inequivalent) definitions in use. Notably in some contexts only the trivialization of the <a class="existingWikiWord" href="/nlab/show/canonical+bundle">canonical bundle</a> is required, but not the vanishing of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mrow><mn>0</mn><mo><</mo><mo>•</mo><mo><</mo><mi>n</mi></mrow></msup><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>𝒪</mi> <mi>X</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">H^{0 \lt \bullet \lt n}(X,\mathcal{O}_X)</annotation></semantics></math>. To be explicit on this one sometimes speaks for emphasis of “strict” CY varieties when including this condition.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>in <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 1: an <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a> is a pointed Calabi-Yau <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>-fold.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> 2: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="in_terms_of_structure">In terms 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>-structure</h3> <p>Calabi-Yau structure is equivalently <a class="existingWikiWord" href="/nlab/show/integrability+of+G-structure">integrable</a> <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">G = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SU%28n%29">SU(n)</a>.</p> <p>Details are in <a href="#Prins16">Prins 16, Prop. 1.3.2</a>. See also <a href="#Vezzoni06">Vezzoni 06, p. 24</a>.</p> <h3 id="InSUBordismTheory">In SU-bordism theory</h3> <p>We discuss the classes of Calabi-Yau manifolds seen in <a class="existingWikiWord" href="/nlab/show/SU-bordism+theory">SU-bordism theory</a>. For more see <em><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds+in+SU-bordism+theory">Calabi-Yau manifolds in SU-bordism theory</a></em>.</p> <div class="num_prop" id="K3SurfaceSpansSUBordismRingInDegree4"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a> spans <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> in degree 4)</strong></p> <p>The degree-4 generator <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>y</mi> <mn>4</mn></msub><mo>∈</mo><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup></mrow><annotation encoding="application/x-tex">y_4 \in \Omega^{SU}_4</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> (Prop. <a class="maruku-ref" href="#SUBordismRingAwayFromTwo"></a>) is represented by minus the class of any (non-<a class="existingWikiWord" href="/nlab/show/torus">torus</a>) <a class="existingWikiWord" href="/nlab/show/K3-surface">K3-surface</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mn>4</mn> <mi>SU</mi></msubsup><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo maxsize="1.2em" minsize="1.2em">[</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><mo maxsize="1.2em" minsize="1.2em">]</mo><mo maxsize="1.2em" minsize="1.2em">⟨</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">[</mo><mi>K</mi><mn>3</mn><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^{SU}_4 \;\simeq\; \mathbb{Z}\big[ \tfrac{1}{2}\big]\big\langle -[K3] \big\rangle \,. </annotation></semantics></math></div></div> <p>(<a href="#LLP17">LLP 17, Lemma 1.5, Example 3.1</a>, <a href="#CLP19">CLP 19, Theorem 13.5a</a>)</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a> generate the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">away from 2</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">away from 2</a> is multiplicatively generated by <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a>.</p> </div> <p>(<a href="#LLP17">LLP 17, Theorem 2.4</a>)</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a> in complex dim <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>≤</mo><mn>4</mn></mrow><annotation encoding="application/x-tex">\leq 4</annotation></semantics></math> generate the <a class="existingWikiWord" href="/nlab/show/SU-bordism+ring">SU-bordism ring</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo>≤</mo><mn>8</mn></mrow><annotation encoding="application/x-tex">deg \leq 8</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/localization+of+a+ring">away from 2</a>)</strong></p> <p>There are <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds">Calabi-Yau manifolds</a> of complex dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>3</mn></mrow><annotation encoding="application/x-tex">3</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>4</mn></mrow><annotation encoding="application/x-tex">4</annotation></semantics></math> whose whose SU-bordism classes equal the generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><msub><mi>y</mi> <mn>6</mn></msub></mrow><annotation encoding="application/x-tex">\pm y_6</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>±</mo><msub><mi>y</mi> <mn>8</mn></msub></mrow><annotation encoding="application/x-tex">\pm y_8</annotation></semantics></math> in Prop. <a class="maruku-ref" href="#SUBordismRingAwayFromTwo"></a>.</p> </div> <p>(<a href="#CLP19">CLP 19, Theorem 13.5b,c</a>)</p> <h3 id="artinmazur_formal_group">Artin-Mazur formal group</h3> <p>Over an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a> of <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-dimensional Calabi-Yau variety <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> has an <a class="existingWikiWord" href="/nlab/show/Artin-Mazur+formal+group">Artin-Mazur formal group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Φ</mi> <mi>X</mi> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\Phi^n_X</annotation></semantics></math> which gives the <a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a> of the trivial <a class="existingWikiWord" href="/nlab/show/line+n-bundle">line n-bundle</a> over <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>.</p> <p>See also (<a href="#GeerKatsura03">Geer-Katsura 03</a>).</p> <h3 id="as_supersymmetric_compactification_spaces_in_string_theory">As supersymmetric compactification spaces in string theory</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry+and+Calabi-Yau+manifolds">supersymmetry and Calabi-Yau manifolds</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory+on+CY3-manifolds">heterotic string theory on CY3-manifolds</a></p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong>classification of <a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a> <a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> by <a class="existingWikiWord" href="/nlab/show/Berger%27s+theorem">Berger's theorem</a>:</strong></p> <table><thead><tr><th></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/special+holonomy">special holonomy</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/dimension">dimension</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math>preserved <a class="existingWikiWord" href="/nlab/show/differential+form">differential form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℂ</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{C}\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+manifold">Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/unitary+group">U(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace><mn>2</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,2n\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/K%C3%A4hler+forms">Kähler forms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\omega_2\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/special+unitary+group">SU(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace><mn>2</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,2n\,</annotation></semantics></math></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>ℍ</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{H}\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/quaternionic+K%C3%A4hler+manifold">quaternionic Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Sp%28n%29.Sp%281%29">Sp(n).Sp(1)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace><mn>4</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,4n\,</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><mspace width="thinmathspace"></mspace><msub><mi>ω</mi> <mn>4</mn></msub><mo>=</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>ω</mi> <mn>3</mn></msub><mo>∧</mo><msub><mi>ω</mi> <mn>3</mn></msub><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\omega_4 = \omega_1\wedge \omega_1+ \omega_2\wedge \omega_2 + \omega_3\wedge \omega_3\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/hyper-K%C3%A4hler+manifold">hyper-Kähler manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/symplectic+group">Sp(n)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace><mn>4</mn><mi>n</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,4n\,</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><mspace width="thinmathspace"></mspace><mi>ω</mi><mo>=</mo><mi>a</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi>b</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo stretchy="false">)</mo></mrow></msubsup><mo>+</mo><mi>c</mi><msubsup><mi>ω</mi> <mn>2</mn> <mrow><mo stretchy="false">(</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></msubsup><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\omega = a \omega^{(1)}_2+ b \omega^{(2)}_2 + c \omega^{(3)}_2\,</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>b</mi> <mn>2</mn></msup><mo>+</mo><msup><mi>c</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">a^2 + b^2 + c^2 = 1</annotation></semantics></math>)</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace><mi>𝕆</mi><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,\mathbb{O}\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Spin%287%29+manifold">Spin(7) manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Spin%287%29">Spin(7)</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math>8<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Cayley+form">Cayley form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G%E2%82%82+manifold">G₂ manifold</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/G%E2%82%82">G₂</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</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><mspace width="thinmathspace"></mspace><mn>7</mn><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,7\,</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><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/associative+3-form">associative 3-form</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></td></tr> </tbody></table> </div> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+object">Calabi-Yau object</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+algebra">Calabi-Yau algebra</a>, <strong>Calabi-Yau manifold</strong>, <a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+cohomology">Calabi-Yau cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+space+of+Calabi-Yau+spaces">moduli space of Calabi-Yau spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/SU-cobordism+theory">SU-cobordism theory</a></p> </li> </ul> <h2 id="references">References</h2> <p>The original articles:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a>, <em>On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampére equation</em>, Communications on Pure and Applied Mathematics, <p><strong>31</strong> 3 (1978) 339-411 [<a href="https://doi.org/10.1002/cpa.3160310304">doi:10.1002/cpa.3160310304</a>]</p> </li> </ul> <p>Surveys and review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Shing-Tung+Yau">Shing-Tung Yau</a>, <em><a href="http://www.scholarpedia.org/article/Calabi-Yau_manifold">Calabi-Yau manifold</a></em>, Scholarpedia <strong>4</strong> 86524 (2009) [<a href="https://doi.org/10.4249/scholarpedia.6524">doi</a>]</li> </ul> <p>Motivated from the relation between <a class="existingWikiWord" href="/nlab/show/supersymmetry+and+Calabi-Yau+manifolds">supersymmetry and Calabi-Yau manifolds</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tristan+H%C3%BCbsch">Tristan Hübsch</a>, <em>Calabi-Yau Manifolds – A Bestiary for Physicists</em>, World Scientific 1992 (<a href="https://doi.org/10.1142/1410">doi:10.1142/1410</a>)</li> </ul> <p>In terms of <a class="existingWikiWord" href="/nlab/show/G-structure">G-structure</a>:</p> <ul> <li id="Vezzoni06"> <p>Luigi Vezzoni, <em>The geometry of some special <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n)</annotation></semantics></math>-structure</em>, 2006 (<a href="https://core.ac.uk/download/pdf/14699671.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/VezzoniSUnStructure.pdf" title="pdf">pdf</a>)</p> </li> <li id="Prins16"> <p><a class="existingWikiWord" href="/nlab/show/Dani%C3%ABl+Prins">Daniël Prins</a>, Section 1.3 of: <em>On flux vacua, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SU(n)</annotation></semantics></math>-structures and generalised complex geometry</em>, Université Claude Bernard – Lyon I, 2015. (<a href="https://arxiv.org/abs/1602.05415">arXiv:1602.05415</a>, <a href="https://tel.archives-ouvertes.fr/tel-01280717">tel:01280717</a>)</p> </li> </ul> <p>Discussion of the case of <a class="existingWikiWord" href="/nlab/show/positive+number">positive</a> <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> includes</p> <ul> <li id="GeerKatsura03"><a class="existingWikiWord" href="/nlab/show/Gerard+van+der+Geer">Gerard van der Geer</a>, T. Katsura, <em>On the height of Calabi-Yau varieties in positive characteristic</em> (<a href="http://arxiv.org/abs/math/0302023">arXiv:math/0302023</a>)</li> </ul> <p>The following page collects information on Calabi-Yau manifolds with an eye to application in <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a> (e.g. <a class="existingWikiWord" href="/nlab/show/supersymmetry+and+Calabi-Yau+manifolds">supersymmetry and Calabi-Yau manifolds</a>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Sheldon+Katz">Sheldon Katz</a>, <a class="existingWikiWord" href="/nlab/show/Rolf+Schimmrigk">Rolf Schimmrigk</a>, Andreas Wißkirchen, <em><a href="http://www.th.physik.uni-bonn.de/th/Supplements/cy.html">CALABI-YAU</a></em></li> </ul> <p>Discussion of the relation between the various shades of definitions includes</p> <ul> <li>MathOverflow: <em><a href="http://mathoverflow.net/q/42707/381">Calabi-Yau manifolds</a></em></li> </ul> <p>Mathematical review of the relation to <a class="existingWikiWord" href="/nlab/show/quiver+representations">quiver representations</a> and <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a> includes</p> <ul> <li>Yang-Hui He, <em>Calabi-Yau Varieties: from Quiver Representations to Dessins d’Enfants</em> (<a href="https://arxiv.org/abs/1611.09398">arXiv:1611.09398</a>)</li> </ul> <p>Discussion of CYs in <a class="existingWikiWord" href="/nlab/show/positive+characteristic">positive characteristic</a> includes</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Philip+Candelas">Philip Candelas</a>, <a class="existingWikiWord" href="/nlab/show/Xenia+de+la+Ossa">Xenia de la Ossa</a>, Fernando Rodriguez-Villegas, <em>Calabi-Yau Manifolds Over Finite Fields, I</em> (<a href="http://arxiv.org/abs/hep-th/0012233">arXiv:hep-th/0012233</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Philip+Candelas">Philip Candelas</a>, <a class="existingWikiWord" href="/nlab/show/Xenia+de+la+Ossa">Xenia de la Ossa</a>, Fernando Rodriguez-Villegas, <em>Calabi-Yau Manifolds Over Finite Fields, II</em> (<a href="http://arxiv.org/abs/hep-th/0402133">arXiv:hep-th/0402133</a>)</p> </li> </ul> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+orbifolds">Calabi-Yau orbifolds</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dominic+Joyce">Dominic Joyce</a>, <em>On the topology of desingularizations of Calabi-Yau orbifolds</em> (<a href="https://arxiv.org/abs/math/9806146">arXiv:math/9806146</a>, <a href="https://inspirehep.net/literature/485280">spire:485280</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dominic+Joyce">Dominic Joyce</a>, <em>Deforming Calabi-Yau orbifolds</em>, Asian Journal of Mathematics 3.4 (1999): 853-868 (<a href="https://dx.doi.org/10.4310/AJM.1999.v3.n4.a7">doi:10.4310/AJM.1999.v3.n4.a7</a> <a href="https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/1999/0003/0004/AJM-1999-0003-0004-a007.pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dominic+Joyce">Dominic Joyce</a>, Section 6.5.1 of: <em>Compact Manifolds with Special Holonomy</em>, Oxford Mathematical Monographs, Oxford University Press (2000) (<a href="https://global.oup.com/academic/product/compact-manifolds-with-special-holonomy-9780198506010?cc=de&lang=en&">ISBN:9780198506010</a>)</p> </li> <li> <p>Wei-Dong Ruan, Yuguang Zhang, <em>Convergence of Calabi-Yau manifolds</em>, Advances in Mathematics Volume 228, Issue 3, 20 October 2011, Pages 1543-1589 (<a href="https://arxiv.org/abs/0905.3424">arXiv:0905.3424</a>, <a href="https://doi.org/10.1016/j.aim.2011.06.023">doi:10.1016/j.aim.2011.06.023</a>)</p> </li> <li> <p>Ronan J. Conlon, Anda Degeratu, Frédéric Rochon, <em>Quasi-asymptotically conical Calabi-Yau manifolds</em>. Geom. Topol. 23 (2019) 29-100 (<a href="https://arxiv.org/abs/1611.04410">arXiv:1611.04410</a>)</p> </li> </ul> <p>and in view of <a class="existingWikiWord" href="/nlab/show/mirror+symmetry">mirror symmetry</a>:</p> <ul> <li> <p>Shi-Shyr Roan, <em>The mirror of Calabi-Yau orbifold</em>, International Journal of Mathematics Vol. 02, No. 04, pp. 439-455 (1991) (<a href="https://doi.org/10.1142/S0129167X91000259">doi:10.1142/S0129167X91000259</a>)</p> </li> <li> <p>Alan Stapledon, <em>New mirror pairs of Calabi-Yau orbifolds</em>, Adv. Math. 230 (2012), no. 4-6, 1557-1596 (<a href="https://arxiv.org/abs/1011.5006">arXiv:1011.5006</a>)</p> </li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifolds+in+SU-bordism+theory">Calabi-Yau manifolds in SU-bordism theory</a>:</p> <ul> <li id="LLP17"> <p><a class="existingWikiWord" href="/nlab/show/Ivan+Limonchenko">Ivan Limonchenko</a>, <a class="existingWikiWord" href="/nlab/show/Zhi+Lu">Zhi Lu</a>, <a class="existingWikiWord" href="/nlab/show/Taras+Panov">Taras Panov</a>, <em>Calabi-Yau hypersurfaces and SU-bordism</em>, Proceedings of the Steklov Institute of Mathematics 302 (2018), 270-278 (<a href="https://arxiv.org/abs/1712.07350">arXiv:1712.07350</a>)</p> </li> <li id="CLP19"> <p>Georgy Chernykh, <a class="existingWikiWord" href="/nlab/show/Ivan+Limonchenko">Ivan Limonchenko</a>, <a class="existingWikiWord" href="/nlab/show/Taras+Panov">Taras Panov</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi></mrow><annotation encoding="application/x-tex">SU</annotation></semantics></math>-bordism: structure results and geometric representatives</em>, Russian Math. Surveys 74 (2019), no. 3, 461-524 (<a href="https://arxiv.org/abs/1903.07178">arXiv:1903.07178</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Taras+Panov">Taras Panov</a>, <em>A geometric view on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SU</mi></mrow><annotation encoding="application/x-tex">SU</annotation></semantics></math>-bordism</em>, talk at Moscow State University 2020 (<a href="https://www.math.princeton.edu/events/geometric-view-su-bordism-2020-09-17t170000">webpage</a>, <a href="http://higeom.math.msu.su/people/taras/talks/2019SPb-Panov.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/PanovSU-Bordism.pdf" title="pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 15, 2024 at 09:52:23. 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