The Kernel Bundle of a Holomorphic Fredholm Family

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