"Compact Kähler manifolds"의 두 판 사이의 차이

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

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 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'}]

@@ 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>
@@ 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.
 [[분류:migrate]]
+==메타데이터==
+===위키데이터===
+* ID :  [https://www.wikidata.org/wiki/Q1353916 Q1353916]
+===Spacy 패턴 목록===
+* [{'LOWER': 'kähler'}, {'LEMMA': 'manifold'}]