"Compact Kähler manifolds"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
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 $h^{p,q}=h^{n-p,n-q}$) |
imported>Pythagoras0 |
||
| 5번째 줄: | 5번째 줄: | ||
* $h^{p,q}=h^{q,p}$ | * $h^{p,q}=h^{q,p}$ | ||
* Serre duality $h^{p,q}=h^{n-p,n-q}$ | * 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 | ||
| + | $$ | ||
| + | H^{m}_{dR}(M;\mathbb{C})=\oplus_{p+q=m}H^{p,q}(M) | ||
| + | $$ | ||
| + | Moreover, $H^{p,q}(M)=\overline{H^{q,p}(M)}$. In other words, $H^{m}_{dR}(M)$ carries a real Hodge structure of weight $m$. | ||
2012년 12월 4일 (화) 15:46 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
$$ H^{m}_{dR}(M;\mathbb{C})=\oplus_{p+q=m}H^{p,q}(M) $$ Moreover, $H^{p,q}(M)=\overline{H^{q,p}(M)}$. In other words, $H^{m}_{dR}(M)$ carries a real Hodge structure of weight $m$.