Compact Kähler manifolds - 편집 역사

2021년 2월 17일 (수) 09:31에 Pythagoras0님의 편집

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Pythagoras0: /* 메타데이터 */ 새 문단

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메타데이터: 새 문단

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imported>Pythagoras0: /* introduction */

2017-04-19T01:38:29Z

introduction

imported>Pythagoras0: section 'expositions' updated

2016-05-20T01:05:53Z

section 'expositions' updated

imported>Pythagoras0: /* articles */

2015-09-14T03:31:34Z

articles

imported>Pythagoras0: /* expositions */

2015-08-25T09:01:47Z

expositions

@@ 81번째 줄: / 81번째 줄: @@
 [[분류:migrate]]
-== 메타데이터 ==
+==메타데이터==
 ===위키데이터===
 * ID :  [https://www.wikidata.org/wiki/Q1353916 Q1353916]

← 이전 판		2020년 12월 28일 (월) 02:29 판
80번째 줄:		80번째 줄:
	* 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.		* 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]]		[[분류:migrate]]
		+
		+	== 메타데이터 ==
		+
		+	===위키데이터===
		+	* ID : [https://www.wikidata.org/wiki/Q1353916 Q1353916]

@@ 2번째 줄: / 2번째 줄: @@
 * {{수학노트|url=Metrics_on_Riemann_surfaces}}
 * Kahler geometry = intersection of Riemmanian, Complex, symplectic geometry
-* A Hermitian metric $h$ on a complex manifold $(M^{2m},J)$ : $h(X,Y)=h(JX,JY)$
+* 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) $(1,1)$-form given by $\Omega=i\sum_{\alpha,\beta=1}^{m}h_{\alpha\overline{\beta}}dz^{\alpha}\wedge dz^{\overline{\beta}}$
+* 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 $\Omega$ is closed, i.e., $d\Omega=0$, we call $h$ a Kahler metric
+* 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 $K$ such that $\Omega=i\partial \overline{\partial} K$, which we call the Kahler potential
+* 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 $h$ be a Hermitian metric and the coefficient
+* 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====
-* $h_{\alpha\overline{\beta}}=\frac{1}{2}\delta_{\alpha\beta}$
+* <math>h_{\alpha\overline{\beta}}=\frac{1}{2}\delta_{\alpha\beta}</math>
-* $\Omega=\frac{i}{2}\sum_{\alpha=1}^m dz_{\alpha}\wedge d\bar{z}_{\alpha}$
+* <math>\Omega=\frac{i}{2}\sum_{\alpha=1}^m dz_{\alpha}\wedge d\bar{z}_{\alpha}</math>
-* potential $u(z)=\frac{1}{2}|z|^2$
+* potential <math>u(z)=\frac{1}{2}|z|^2</math>
 ====dimension 1 case====
-* $h_{\alpha\overline{\alpha}}=h_{\overline{\alpha}\alpha}:=h$
+* <math>h_{\alpha\overline{\alpha}}=h_{\overline{\alpha}\alpha}:=h</math>
-* $\Omega=-2ih\,dz \wedge d\overline{z}$
+* <math>\Omega=-2ih\,dz \wedge d\overline{z}</math>
-* for $\mathbb{P}^{1}$,
+* 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]]
@@ 41번째 줄: / 41번째 줄: @@
 * compact Kähler manifold of dimension n
 * Dolbeault cohomology
-* $h^{p,q}=\operatorname{dim} H^{p,q}(X)$
+* <math>h^{p,q}=\operatorname{dim} H^{p,q}(X)</math>
-* $h^{p,q}=h^{q,p}$
+* <math>h^{p,q}=h^{q,p}</math>
-* Serre duality $h^{p,q}=h^{n-p,n-q}$
+* Serre duality <math>h^{p,q}=h^{n-p,n-q}</math>
 ===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
+* 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, $H^{p,q}(M)=\overline{H^{q,p}(M)}$. In other words, $H^{m}_{dR}(M)$ carries a real Hodge structure of weight $m$.
+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>.
@@ 57번째 줄: / 57번째 줄: @@
 * cohomology of sheaves of holomorphic forms
 ;theorem
-Let $\Omega$ be the space of holomorphic $p$-forms on $M$
+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>

← 이전 판	2020년 11월 13일 (금) 17:29 판
1번째 줄:	1번째 줄:
	+	==introduction==
	+	* {{수학노트\|url=Metrics_on_Riemann_surfaces}}
	+	* 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====
	+	* [[Fubini–Study metric]]
	+	* [[K3 surfaces]]
	+	* [[Calabi-Yau manifold]]
	+	* [[Hyperkahler manifolds]]
	+
	+
	+	==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==
	+	* 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]]

← 이전 판		2020년 11월 13일 (금) 17:29 판
1번째 줄:		1번째 줄:
−	~~==introduction==~~
−	* {{수학노트\|url=Metrics_on_Riemann_surfaces}}
−	* 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====~~
−	* [[Fubini–Study metric]]
−	* [[K3 surfaces]]
−	* [[Calabi-Yau manifold]]
−	* [[Hyperkahler manifolds]]
−
−
−	~~==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==~~
−	* 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.

← 이전 판	2020년 11월 16일 (월) 17:54 판
(차이 없음)

@@ 68번째 줄: / 68번째 줄: @@
 ==expositions==
 * 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.

@@ 75번째 줄: / 75번째 줄: @@
 ==articles==
 * 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.