The Kernel Bundle of a Holomorphic Fredholm Family

Abstract

Let Y be a smooth connected manifold, Σ ⊂ C an open set and (σ, y) → Py(σ) a family of unbounded Fredholm operators D ⊂ H 1 → H 2 of index 0 depending smoothly on (y, σ) ∈ Y × Σ and holomorphically on σ. We show how to associate to P, under mild hypotheses, a smooth vector bundle K → Y whose fiber over a given y ∈ Y consists of classes, modulo holomorphic elements, of meromorphic elements φ with Pyφ holomorphic. As applications we give two examples relevant in the general theory of boundary value problems for elliptic wedge operators.2010 Mathematics Subject Classification. Primary: 58J32; Secondary: 58J05,35J48,35J58.

Hodge theory on Cheeger spaces

“…This follows from Theorem 5.6, which identifies these Hodge cohomology spaces with de Rham cohomology spaces and Theorem 5.10 which shows that the latter are invariant under pull-back by stratified diffeomorphism. This also follows by combining Theorem 4.4, which shows that Assumption 3 holds, with [KM13].…”

“…A refined analysis of more general boundary value problem of this type on spaces with simple edge singularities (i.e. on depth 1 spaces) is carried out in [MV], and in a series of articles by Krainer and Mendoza, [KM13,KM15,KM13a,GKM].…”

“…Quite a lot can still be done even without Assumption 2. For example, in a series of papers [KM13,KM15,KM13a] and further ongoing work, Krainer and Mendoza study elliptic edge operators assuming only that spec b (P ) does not intersect the boundaries of this same strip, but allowing the indicial data in this range to vary.…”

Boundary Value Problems for Elliptic Wedge Operators: The First-Order Case

“…In [10] we also prove that if A ⋆ is the formal adjoint of A and T ⋆ its trace bundle, then, taking γ = m/2 for convenience,…”

“…Briefly, in [10] we address the fundamental issue of boundary values, in [11] we construct an extension of the Douglis-Nirenberg calculus (see for instance [3,9] for the role of this calculus in the classical context), while in [12] we address elliptic boundary value problems for first order wedge operators and prove, in particular, sufficient conditions for well-posedness of such problems. Here we shall address the main aspects of each of these papers in subsequent sections, after dealing with background information and some notation.…”