$ℂ$ as a Kähler manifold

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

Parametrize ${ℝ}^2$ 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 ${\mathrm{span}}_{ℝ}\{\frac{\partial }{\partial x},\frac{\partial }{\partial y}\}$ and the complex structure (http://planetmath.org/AlmostComplexStructure) $J$ is defined by

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

$J\left({\displaystyle \frac{\partial }{\partial y}}\right)=-{\displaystyle \frac{\partial }{\partial x}}$(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 $(x,y)$ coordinates.

So lets verify all the points in the definition.

• $ℂ$ is a Riemannian Manifold

• $g$ is Hermitian.

  $$g(J\frac{\partial }{\partial x},J\frac{\partial }{\partial y})=g(\frac{\partial }{\partial y},-\frac{\partial }{\partial x})=0=g(\frac{\partial }{\partial x},\frac{\partial }{\partial y})$$

  $$g(J\frac{\partial }{\partial x},J\frac{\partial }{\partial x})=g(\frac{\partial }{\partial y},\frac{\partial }{\partial y})=1=g(\frac{\partial }{\partial x},\frac{\partial }{\partial x})$$

  $$g(J\frac{\partial }{\partial y},J\frac{\partial }{\partial y})=g(-\frac{\partial }{\partial x},-\frac{\partial }{\partial x})=1=g(\frac{\partial }{\partial y},\frac{\partial }{\partial y})$$

• $J$ 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.

$ℂ$ 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 ${ℝ}^n$ seen as a metric space.