"Compact Kähler manifolds"의 두 판 사이의 차이
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imported>Pythagoras0 (새 문서: ==cohomology theory== * compact Kähler manifold of dimension n * Dolbeault cohomology * $h^{p,q}=\operatorname{dim} H^{p,q}(X)$ * $h^{p,q}=h^{q,p}$ * Serre duality $h^{p,q}=h^{n-p,n-q}$) |
Pythagoras0 (토론 | 기여) |
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(사용자 2명의 중간 판 35개는 보이지 않습니다) | |||
1번째 줄: | 1번째 줄: | ||
+ | ==introduction== | ||
+ | * {{수학노트|url=Metrics_on_Riemann_surfaces}} | ||
+ | * Kahler geometry = intersection of Riemmanian, Complex, symplectic geometry | ||
+ | * A Hermitian metric <math>h</math> on a complex manifold <math>(M^{2m},J)</math> : <math>h(X,Y)=h(JX,JY)</math> | ||
+ | * fundamental 2-form (or Kähler form) <math>(1,1)</math>-form given by <math>\Omega=i\sum_{\alpha,\beta=1}^{m}h_{\alpha\overline{\beta}}dz^{\alpha}\wedge dz^{\overline{\beta}}</math> | ||
+ | * If <math>\Omega</math> is closed, i.e., <math>d\Omega=0</math>, we call <math>h</math> a Kahler metric | ||
+ | * there exists a real function <math>K</math> such that <math>\Omega=i\partial \overline{\partial} K</math>, which we call the Kahler potential | ||
+ | * The Ricci form is one of the most important objects on a Kahler manifold | ||
+ | |||
+ | ==Hermitian metric on a complex manifold== | ||
+ | * Let <math>h</math> be a Hermitian metric and the coefficient | ||
+ | :<math> | ||
+ | h_{\alpha\overline{\beta}}:=h(\frac{\partial}{\partial z_{\alpha}},\frac{\partial}{\partial \overline{z}_{\beta}}) | ||
+ | </math> | ||
+ | |||
+ | |||
+ | ==examples== | ||
+ | ====flat matric==== | ||
+ | * <math>h_{\alpha\overline{\beta}}=\frac{1}{2}\delta_{\alpha\beta}</math> | ||
+ | * <math>\Omega=\frac{i}{2}\sum_{\alpha=1}^m dz_{\alpha}\wedge d\bar{z}_{\alpha}</math> | ||
+ | * potential <math>u(z)=\frac{1}{2}|z|^2</math> | ||
+ | |||
+ | ====dimension 1 case==== | ||
+ | * <math>h_{\alpha\overline{\alpha}}=h_{\overline{\alpha}\alpha}:=h</math> | ||
+ | * <math>\Omega=-2ih\,dz \wedge d\overline{z}</math> | ||
+ | * for <math>\mathbb{P}^{1}</math>, | ||
+ | :<math> | ||
+ | \Omega=\frac{-i}{2\pi}\frac{dz \wedge d\bar{z}}{(1+|z|^2)^2} | ||
+ | </math> | ||
+ | see [[Chern class]] | ||
+ | |||
+ | ====etc==== | ||
+ | * [[Fubini–Study metric]] | ||
+ | * [[K3 surfaces]] | ||
+ | * [[Calabi-Yau manifold]] | ||
+ | * [[Hyperkahler manifolds]] | ||
+ | |||
+ | |||
==cohomology theory== | ==cohomology theory== | ||
+ | * [[Hodge theory of harmonic forms]] | ||
* compact Kähler manifold of dimension n | * compact Kähler manifold of dimension n | ||
* Dolbeault cohomology | * Dolbeault cohomology | ||
− | * | + | * <math>h^{p,q}=\operatorname{dim} H^{p,q}(X)</math> |
− | * | + | * <math>h^{p,q}=h^{q,p}</math> |
− | * Serre duality | + | * Serre duality <math>h^{p,q}=h^{n-p,n-q}</math> |
+ | |||
+ | |||
+ | ===Hodge decomposition theorem=== | ||
+ | * Let <math>M</math> be a compact Kähler manifold. Let <math>H^{p,q}(M)</math> be the space of cohomology classes represented by a closed form of type <math>(p,q)</math>. There is a direct sum decomposition | ||
+ | :<math> | ||
+ | H^{m}_{dR}(M;\mathbb{C})=\bigoplus_{p+q=m}H^{p,q}(M) | ||
+ | </math> | ||
+ | Moreover, <math>H^{p,q}(M)=\overline{H^{q,p}(M)}</math>. In other words, <math>H^{m}_{dR}(M)</math> carries a real Hodge structure of weight <math>m</math>. | ||
+ | |||
+ | |||
+ | ===Delbeault=== | ||
+ | * cohomology of sheaves of holomorphic forms | ||
+ | ;theorem | ||
+ | Let <math>\Omega</math> be the space of holomorphic <math>p</math>-forms on <math>M</math> | ||
+ | :<math> | ||
+ | H^{p,q}(M)\cong H^q(M,\Omega^p) | ||
+ | </math> | ||
+ | |||
+ | |||
+ | ==computational resource== | ||
+ | * https://docs.google.com/file/d/0B8XXo8Tve1cxLXgxazdpTXRDR0E/edit | ||
+ | |||
+ | |||
+ | ==expositions== | ||
+ | * Valentino Tosatti, KAWA lecture notes on the Kähler-Ricci flow, arXiv:1508.04823 [math.DG], August 19 2015, http://arxiv.org/abs/1508.04823 | ||
+ | * Tosatti, Valentino. “Uniqueness of CP^n.” arXiv:1508.05641 [math], August 23, 2015. http://arxiv.org/abs/1508.05641. | ||
+ | * Weinkove, Ben. ‘The K"ahler-Ricci Flow on Compact K"ahler Manifolds’. arXiv:1502.06855 [math], 24 February 2015. http://arxiv.org/abs/1502.06855. | ||
+ | * Stefan Vandoren [http://www.staff.science.uu.nl/~vando101/MRIlectures.pdf Lectures on Riemannian Geometry, Part II:Complex Manifolds] | ||
+ | * [http://www.math.upenn.edu/~siegelch/Notes/Cattani1.pdf Complex manifolds, Kahler metrics, differential and harmonic forms] | ||
+ | * Werner Ballmann [http://people.mpim-bonn.mpg.de/hwbllmnn/archiv/kaehler0609.pdf Lectures on Kahler Manifolds] | ||
+ | |||
+ | ==articles== | ||
+ | * Berczi, Gergely. “Towards the Green-Griffiths-Lang Conjecture via Equivariant Localisation.” arXiv:1509.03406 [math], September 11, 2015. http://arxiv.org/abs/1509.03406. | ||
+ | * Treger, Robert. ‘On Uniformization of Compact Kahler Manifolds’. arXiv:1507.01379 [math], 6 July 2015. http://arxiv.org/abs/1507.01379. | ||
+ | * Dinh, Tien-Cuong, Fei Hu, and De-Qi Zhang. ‘Compact K"ahler Manifolds Admitting Large Solvable Groups of Automorphisms’. arXiv:1502.07060 [math], 25 February 2015. http://arxiv.org/abs/1502.07060. | ||
+ | [[분류:migrate]] | ||
+ | |||
+ | ==메타데이터== | ||
+ | ===위키데이터=== | ||
+ | * ID : [https://www.wikidata.org/wiki/Q1353916 Q1353916] | ||
+ | ===Spacy 패턴 목록=== | ||
+ | * [{'LOWER': 'kähler'}, {'LEMMA': 'manifold'}] |
2021년 2월 17일 (수) 01:31 기준 최신판
introduction
- 틀:수학노트
- Kahler geometry = intersection of Riemmanian, Complex, symplectic geometry
- A Hermitian metric \(h\) on a complex manifold \((M^{2m},J)\) \[h(X,Y)=h(JX,JY)\]
- fundamental 2-form (or Kähler form) \((1,1)\)-form given by \(\Omega=i\sum_{\alpha,\beta=1}^{m}h_{\alpha\overline{\beta}}dz^{\alpha}\wedge dz^{\overline{\beta}}\)
- If \(\Omega\) is closed, i.e., \(d\Omega=0\), we call \(h\) a Kahler metric
- there exists a real function \(K\) such that \(\Omega=i\partial \overline{\partial} K\), which we call the Kahler potential
- The Ricci form is one of the most important objects on a Kahler manifold
Hermitian metric on a complex manifold
- Let \(h\) be a Hermitian metric and the coefficient
\[ h_{\alpha\overline{\beta}}:=h(\frac{\partial}{\partial z_{\alpha}},\frac{\partial}{\partial \overline{z}_{\beta}}) \]
examples
flat matric
- \(h_{\alpha\overline{\beta}}=\frac{1}{2}\delta_{\alpha\beta}\)
- \(\Omega=\frac{i}{2}\sum_{\alpha=1}^m dz_{\alpha}\wedge d\bar{z}_{\alpha}\)
- potential \(u(z)=\frac{1}{2}|z|^2\)
dimension 1 case
- \(h_{\alpha\overline{\alpha}}=h_{\overline{\alpha}\alpha}:=h\)
- \(\Omega=-2ih\,dz \wedge d\overline{z}\)
- for \(\mathbb{P}^{1}\),
\[ \Omega=\frac{-i}{2\pi}\frac{dz \wedge d\bar{z}}{(1+|z|^2)^2} \] see Chern class
etc
cohomology theory
- Hodge theory of harmonic forms
- compact Kähler manifold of dimension n
- Dolbeault cohomology
- \(h^{p,q}=\operatorname{dim} H^{p,q}(X)\)
- \(h^{p,q}=h^{q,p}\)
- Serre duality \(h^{p,q}=h^{n-p,n-q}\)
Hodge decomposition theorem
- Let \(M\) be a compact Kähler manifold. Let \(H^{p,q}(M)\) be the space of cohomology classes represented by a closed form of type \((p,q)\). There is a direct sum decomposition
\[ H^{m}_{dR}(M;\mathbb{C})=\bigoplus_{p+q=m}H^{p,q}(M) \] Moreover, \(H^{p,q}(M)=\overline{H^{q,p}(M)}\). In other words, \(H^{m}_{dR}(M)\) carries a real Hodge structure of weight \(m\).
Delbeault
- cohomology of sheaves of holomorphic forms
- theorem
Let \(\Omega\) be the space of holomorphic \(p\)-forms on \(M\) \[ H^{p,q}(M)\cong H^q(M,\Omega^p) \]
computational resource
expositions
- Valentino Tosatti, KAWA lecture notes on the Kähler-Ricci flow, arXiv:1508.04823 [math.DG], August 19 2015, http://arxiv.org/abs/1508.04823
- Tosatti, Valentino. “Uniqueness of CP^n.” arXiv:1508.05641 [math], August 23, 2015. http://arxiv.org/abs/1508.05641.
- Weinkove, Ben. ‘The K"ahler-Ricci Flow on Compact K"ahler Manifolds’. arXiv:1502.06855 [math], 24 February 2015. http://arxiv.org/abs/1502.06855.
- Stefan Vandoren Lectures on Riemannian Geometry, Part II:Complex Manifolds
- Complex manifolds, Kahler metrics, differential and harmonic forms
- Werner Ballmann Lectures on Kahler Manifolds
articles
- Berczi, Gergely. “Towards the Green-Griffiths-Lang Conjecture via Equivariant Localisation.” arXiv:1509.03406 [math], September 11, 2015. http://arxiv.org/abs/1509.03406.
- Treger, Robert. ‘On Uniformization of Compact Kahler Manifolds’. arXiv:1507.01379 [math], 6 July 2015. http://arxiv.org/abs/1507.01379.
- Dinh, Tien-Cuong, Fei Hu, and De-Qi Zhang. ‘Compact K"ahler Manifolds Admitting Large Solvable Groups of Automorphisms’. arXiv:1502.07060 [math], 25 February 2015. http://arxiv.org/abs/1502.07060.
메타데이터
위키데이터
- ID : Q1353916
Spacy 패턴 목록
- [{'LOWER': 'kähler'}, {'LEMMA': 'manifold'}]