The kernel bundle of a holomorphic Fredholm family

Let $\text{Y}$ be a smooth connected manifold, $\mathrm{\Sigma }\subset \text{C}$ an open set and $(\sigma ,y)\to {\text{scrP}}_y(\sigma )$ a family of unbounded Fredholm operators $D\subset H_1\to H_2$ of index 0 depending smoothly on $(y,\sigma)\in \Y\times \Sigma$andholomorphicallyon$\sigma$\mathrm{.}Weshowhowtoassociateto$\scrP$, under mild hypotheses, a smooth vector bundle $\text{kerb}\to \text{Y}$ whose fiber over a given $y\in \text{Y}$ consists of classes, modulo holomorphic elements, of meromorphic elements $\varphi$ with ${\text{scrP}}_y\varphi$ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.