"Compact Kähler manifolds"의 두 판 사이의 차이
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imported>Pythagoras0 |
imported>Pythagoras0 |
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| 10번째 줄: | 10번째 줄: | ||
* 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 $M$ 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 | ||
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| − | H^{m}_{dR}(M;\mathbb{C})=\ | + | H^{m}_{dR}(M;\mathbb{C})=\bigoplus_{p+q=m}H^{p,q}(M) |
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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, $H^{p,q}(M)=\overline{H^{q,p}(M)}$. In other words, $H^{m}_{dR}(M)$ carries a real Hodge structure of weight $m$. | ||
2012년 12월 4일 (화) 15:47 판
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 $m$.