Here the complex structure does not play a special role. It is true more generally that, for an open subset $\Omega \subset \mathbb{R}^m$ and a $C^r$ family of $n\times n$ real matrices $\mathbf{A}:\Omega\to \mathrm{M} _ n (\mathbb{R})$ with constant rank, the distribution $\operatorname{ker}\mathbf{A}(x)$ is a $C^r$ distribution.

Since the matrices $\mathbf{A}(x)$ have constant rank, for any $x_0\in\Omega$ there is nbd $U$ of $x_0$ and a number $\epsilon > 0$ such that $0$ is the only eigenvalue of $\mathbf{A}(x)$ in $\overline {B(0,\epsilon)}\subset\mathbb{C}$ for all $x\in U$. Therefore the spectral projector of the eigenvalue $0$ of $\mathbf{A}(x)$ represents as an integral along the same path that bounds $ B(0,\epsilon)$, for all $x\in U$ $$P(x)=-\frac{1}{2\pi i}\int_{\Gamma}\big( \mathbf{A}(x) - \lambda\big)^{-1}d \lambda \, ,$$ which is enough to show the $C^r$ dependence of $\operatorname{ker}\mathbf{A}(x)$.

For other more delicate matters I recommend you to check Kato's Perturbation Theory for Linear Operators.

Here the complex structure does not play a special role. It is true more generally that, for an open subset $\Omega \subset \mathbb{R}^m$ and a $C^r$ family of $n\times n$ real matrices $\mathbf{A}:\Omega\to \mathrm{M} _ n (\mathbb{R})$ with constant rank, the distribution $\operatorname{ker}\mathbf{A}(x)$ is a $C^r$ distribution.

From the assumption $\operatorname {dim}\operatorname{ker}\mathbf{A}(x)$ is also constant, and for any $x_0\in\Omega$ there is nbd $U$ of $x_0$ and a number $\epsilon > 0$ such that $0$ is the only eigenvalue of $\mathbf{A}(x)$ in $\overline {B(0,\epsilon)}\subset\mathbb{C}$ for all $x\in U$. Therefore the spectral projector of the eigenvalue $0$ of $\mathbf{A}(x)$ represents as an integral along the same path that bounds $ B(0,\epsilon)$, for all $x\in U$ $$P(x)=-\frac{1}{2\pi i}\int_{\Gamma}\big( \mathbf{A}(x) - \lambda\big)^{-1}d \lambda \, ,$$ which is enough to show the $C^r$ dependence of $\operatorname{ker}\mathbf{A}(x)$.

For other more delicate matters I recommend you to check Kato's Perturbation Theory for Linear Operators.