"Compact Kähler manifolds"의 두 판 사이의 차이

2013년 6월 3일 (월) 23:06 판

introduction

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 $h$ Kahler
there exists a real function $u$ such that $\Omega=i\partial \overline{\partial} u$, 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

$$ h_{\alpha\overline{\beta}}:=h(\frac{\partial}{\partial z_{\alpha}},\frac{\partial}{\partial \overline{z}_{\beta}}) $$

dimension 1 case

$h_{\alpha\overline{\alpha}}=h_{\overline{\alpha}\alpha}:=h$
$\omega=-2ih\,dz d\overline{z}$
for $\mathbb{P}^{1}$,

$$ \omega=\frac{-i}{2\pi}\frac{dz \wedge d\bar{z}}{(1+|z|^2)^2} $$ see Chern class

examples

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 $h^{p,q}=h^{n-p,n-q}$

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 $m$.

@@ 1번째 줄: / 1번째 줄: @@
 ==introduction==
-* Hermitian complex manifold $M$ equipped with a closed Kähler from $\omega$, i.e., $d\omega=0$
+* A Hermitian metric $h$ on a complex manifold $(M^{2m},J)$ : $h(X,Y)=h(JX,JY)$
-* $\omega=-2ih_{\alpha\overline{\beta}}dz^{\alpha}dz^{\overline{\beta}}$
+* 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 $h$ Kahler
+* there exists a real function $u$ such that $\Omega=i\partial \overline{\partial} u$, which we call the Kahler potential
 * The Ricci form is one of the most important objects on a Kahler manifold
@@ 10번째 줄: / 12번째 줄: @@
 h_{\alpha\overline{\beta}}:=h(\frac{\partial}{\partial z_{\alpha}},\frac{\partial}{\partial \overline{z}_{\beta}})
 $$
 ==dimension 1 case==
@@ 16번째 줄: / 19번째 줄: @@
 * for $\mathbb{P}^{1}$,
 $$
-\omega=\frac{-i}{2\pi}\frac{dz d\bar{z}}{(1+|z|^2)^2}
+\omega=\frac{-i}{2\pi}\frac{dz \wedge d\bar{z}}{(1+|z|^2)^2}
 $$
 see [[Chern class]]