The Kernel Bundle of a Holomorphic Fredholm Family

Let be a smooth connected manifold, Σ ⊂ ℂ an open set and (σ, y) → y (σ) a family of unbounded Fredholm operators D ⊂ H 1 → H 2 of index 0 depending smoothly on (y, σ) ∈ × Σ and holomorphically on σ. We show how to associate to , under mild hypotheses, a smooth vector bundle → whose fiber over a given y ∈ consists of classes, modulo holomorphic elements, of meromorphic elements φ with y φ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.