How a matrix of C^1 functions on a domain Ω in Cn generates a C^1 distribution in Ω

Question

I am reading the paper Complex foliation generated by (1,1)-forms by M. Klimek. I have a problem understanding why is true a detail in one of the theorems:

Let $\Omega$ be an open connected subset of $\mathbb{C}^n$ ($n$-tuples of complex numbers) and let $\mathbf{A}(z)$ be a $n\times n$ complex matrix for every $z$ in $\Omega$. The entries of the matrix are $C^1$ functions of $z$. For every $z$ in $\Omega$, the matrix $\mathbf{A}(z)$ is reduced by its range i.e the subspaces $\operatorname{Range}(\mathbf{A}(z))$ and its orthogonal complement are invariant subspaces of $\mathbf{A}(z)$.

Thm. If $\operatorname {dim}\operatorname{Range}(\mathbf{A}(z))=p$ with $1 \le p < n$ for every $z$ in $\Omega$ then the family of subspaces $\operatorname{Ker}(\mathbf{A}(z))$ is a $C^1$-distribution of real dimension $2(n-p)$.

The only thing that it is not clear for me is why the subspaces $\operatorname{Ker}(\mathbf{A}(z))$ vary in a $C^1$ way, i.e how to construct at every point a local $C^1$ frame for the distribution.

I understand that a basis for $\operatorname{Ker}(\mathbf{A}(z))$ is make with eigenvectors for the zero eigenvalue but how do we know that these eigenvectors depends on the parameter $z$ in a $C^1$ way?

Thanks

Answer 1

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.