Compact Kähler manifolds

introduction

• Hermitian complex manifold $M$ equipped with a closed Kähler from $\omega$, i.e., $d\omega=0$
• $\omega=-2ih_{\alpha\overline{\beta}}dz^{\alpha}dz^{\overline{\beta}}$
• 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 $\mathbb{P}^{1}$,

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

examples

• complex projective line
• K3 surfaces
• Calabi-Yau manifold

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 $m$.