Compact Kähler manifolds

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 $h$ a Kahler metric
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 $m$.