Kähler coordinate-patch data for a rigid $N=1$ chiral model

Let $n$ be a positive integer. Give ${\mathbb{C}}^n$ the Euclidean topology transported from ${\mathbb{R}}^{2n}$ by the real coordinates $z^i=x^i+i{y}^i$, where $x^i,y^i\in \mathbb{R}$. Let $U\subseteq {\mathbb{C}}^n$ be open in this topology and contain $0$. Each complex number has its unique real decomposition $a+ib$ with $a,b\in \mathbb{R}$; write $\overline{a+ib}=a-ib$. A function on $U$ is smooth if all of its partial derivatives of every finite order in the real coordinates $(x^1,y^1,\dots ,x^n,y^n)$ exist and are continuous. Define the Wirtinger derivatives by ${\mathrm{\partial }}_i=\frac{1}{2}({\mathrm{\partial }}_{x^i}-i{\mathrm{\partial }}_{y^i})$ and ${\mathrm{\partial }}_{\bar{i}}=\frac{1}{2}({\mathrm{\partial }}_{x^i}+i{\mathrm{\partial }}_{y^i})$.

A Kähler coordinate-patch datum is such a pair $(U,K)$ with a smooth real-valued function $K:U\to \mathbb{R}$ for which the matrix $g_{i\bar{j}}(z,\bar{z}):={\mathrm{\partial }}_i{\mathrm{\partial }}_{\bar{j}}K(z,\bar{z})$ satisfies $g_{i\bar{j}}=\overline{g_{j\stackrel{ˉ}{i}}}$ and is positive definite: for every $z\in U$ and every nonzero $v=(v^1,\dots ,v^n)\in {\mathbb{C}}^n$, ${\bar{v}}^{\bar{j}}{g}_{i\bar{j}}(z,\bar{z})v^i>0$. This smooth Hermitian matrix is the Kähler metric specified by the potential $K$. In natural units $\mathrm{\hslash }=c=1$, mass dimensions are powers of mass; assign $[z^i]=1$, $[K]=2$, and $[g_{i\bar{j}}]=0$.

The complex numbers as $\mathbb{R}[x]\mathrm{/}(x^2+1)$, with the real embedding and imaginary unit $i$

$\mathbb{C}=\mathbb{R}[x]\mathrm{/}(x^2+1)$ is a field, every element is uniquely $a+bi$, and every nonzero element has inverse $(a-bi)\mathrm{/}(a^2+b^2)$

Adel Bilal, Introduction to Supersymmetry, §7.1.