# $\mathbb{C}$ as a Kähler manifold

$\mathbb{C}$ can be interpreted as $\mathbb{R}^{2}$ with a complex structure (http://planetmath.org/AlmostComplexStructure) $J$.

Parametrize $\mathbb{R}^{2}$ via the usual coordinates $(x,y)$.

A point $z$ in the complex plane can thus be written $z=x+iy$.

The tangent space at each point is generated by the $\mathrm{span}_{\mathbb{R}}\left\{\frac{\partial}{\partial x},\frac{\partial}{% \partial y}\right\}$ and the complex structure (http://planetmath.org/AlmostComplexStructure) $J$ is defined by11notice $J$ acts as a counterclockwise rotation by $\frac{\pi}{2}$, just as expected

 $\displaystyle J\left(\frac{\partial}{\partial x}\right)=\frac{\partial}{% \partial y}$ (1) $\displaystyle J\left(\frac{\partial}{\partial y}\right)=-\frac{\partial}{% \partial x}$ (2)

The metric can be the usual metric $g=dx\otimes dx+dy\otimes dy$. This is a flat metric and therefore all the covariant derivatives are plain partial derivatives in the $(x,y)$ coordinates22the Christoffel symbols on these coordinates vanish.

So lets verify all the points in the definition.

• $\mathbb{C}$ is a Riemannian Manifold

• $g$ is Hermitian.

 $g\left(J\frac{\partial}{\partial x},J\frac{\partial}{\partial y}\right)=g\left% (\frac{\partial}{\partial y},-\frac{\partial}{\partial x}\right)=0=g\left(% \frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)\\$
 $g\left(J\frac{\partial}{\partial x},J\frac{\partial}{\partial x}\right)=g\left% (\frac{\partial}{\partial y},\frac{\partial}{\partial y}\right)=1=g\left(\frac% {\partial}{\partial x},\frac{\partial}{\partial x}\right)$
 $g\left(J\frac{\partial}{\partial y},J\frac{\partial}{\partial y}\right)=g\left% (-\frac{\partial}{\partial x},-\frac{\partial}{\partial x}\right)=1=g\left(% \frac{\partial}{\partial y},\frac{\partial}{\partial y}\right)$
• $J$ is covariantly constant because its components in the $(x,y)$ coordinates are constant and as previously stated, the covariant derivatives are just partial derivatives in this example.

$\mathbb{C}$ is therefore a Kähler manifold.

The symplectic form for this example is

 $\omega=dx\wedge dy$

This is the simplest example of a Kähler manifold and can be seen as a template for other less trivial examples. Those are generalizations of this example just as Riemannian manifolds are generalizations of $\mathbb{R}^{n}$ seen as a metric space.

Title $\mathbb{C}$ as a Kähler manifold mathbbCAsAKahlerManifold 2013-03-22 15:46:32 2013-03-22 15:46:32 cvalente (11260) cvalente (11260) 16 cvalente (11260) Example msc 53D99 KahlerManifold AlmostComplexStructure SymplecticManifold