Every gluing of complex analytic space germs is large

Prove that for any germs (X,x), (Y,y), and (Z,z) of complex analytic spaces with surjective homomorphisms O_{X,x} → O_{Z,z} and O_{Y,y} → O_{Z,z}, the canonical map f: (X,x) → (X,x) ⊔_{(Z,z)} (Y,y) is a large morphism. Concretely, establish that for every subspace (W,w) of (X,x), the Poincaré series satisfy P_{(W,w)}^{(X,x) ⊔_{(Z,z)} (Y,y)}(t) = P_{(W,w)}^{(X,x)}(t) · P_{(X,x)}^{(X,x) ⊔_{(Z,z)} (Y,y)}(t).

Background

The paper introduces three classes of gluings of complex analytic space germs—weakly large, large, and strongly large—defined via factorization properties of Poincaré series associated to canonical maps from the components into the gluing. A gluing is called large if the canonical map f: (X,x) → (X,x) ⊔{(Z,z)} (Y,y) is a large morphism in the sense of Levin: for every subspace (W,w) of (X,x), the Poincaré series factorizes as P{(W,w)}^{target} = P_{(W,w)}^{source} * P_{source}^{target}.

The authors prove this factorization for broad classes (e.g., when (Z,z) is a reduced point, or under certain weak complete intersection conditions) and derive explicit Poincaré series and Betti number formulas. They conjecture that the factorization holds for every gluing arising from surjective maps to (Z,z), which, if true, would unify and greatly simplify the analysis of invariants of glued analytic germs.