Complex Analytic and Differential Geometry 2025, 26 - q-Convex Spaces

For noetherian local rings R, with unique maximal ideal m, finitely generated R-modules E with mE = E imply E = {0}. If F is another such R-module with submodule E, there is an integer s with E ∩ m^kF = m^k-s(E ∩ m²F) for k ≥ s. Also, ⋒_k≥0 m^kF = {0} and ⋒_k≥0(E + m^kF) = E. For R = O_n with m = (z_i), O_n/m^k is a finite-dim vector space generated by the monomials of m. E/m^kE is a finite-dim vector space for any finitely generated O_n-module E. As ⋒m^kE = {0} there is an injection of E into Π_k∈ℕE/m^kE. If E is Hausdorff of this product, it's said to have Krull topology. If E ⊂ F is finitely generated O_n-modules, then a map F → G = F/E is open and if E is closed in F, and the topology induced by F on E coincides with the Krull topology of E.

For E, F is Hausdorff and locally convex with g: E → F continuous, g is compact if there is a neighborhood U of 0 in E with the closure of g(U) compact in F. g is a monomorphism if g is a topological isomorphism of E onto a closed subspace of F, and a quasi-monomorphism if ker g is finite-dim and h: E/ker g → F a monomorphism. g is an epimorphism if g is surjective and open, and quasi-epimorphic if g is an epimorphism of E onto a closed finite-codim subspace of F. g is quasi-isomorphic if g is simultaneously a quasi-monomorphism and a quasi-epimorphism. If E, F are Frechet spaces, then g is (quasi-)monomorphic iff g(E) is closed in F and g is injective (and ker g is finite-dim). g would be (quasi-)epimorphic iff g is surjective (g(E) is finite-codim.). For h: E → F is compact, and g: E → F is a quasi-monomorphic, then g + h is quasi-monomorphic. If E, F are Frechet, and if g: E → F is quasi-epimorphic, then g + h is a quasi-epimorphic. Let (E^○, d), (F^○, δ) complexes of Frechet spaces with continuous differentials, ρ^○: E^○ → F^○ a continuous complex morphism, ρ^q compact and H^q(ρ^○): H^q(E^○) → H^q(F^○) surjective, then H^q(F^○) is Hausdorff and finite-dim.

If v∈C²(M, ℝ) is strongly/weakly q-convex and Y a submanifold of M, then v_↑Y is strongly(weakly) q-convex. If v is part of an s-long series of q_j-convex functions and χ a increasing/strictly increasing convex function in all variables, then χ(v₁....) is weakly/strongly q-convex with q - 2 = Σ(q_j - 1). The sum of v_i is weakly/strongly q-convex. An analytic scheme (X, O_X) and v a function on X, v is strongly (weakly) q-convex of class C^k on X if it can be covered by patches G: U → A, A⊂Ω⊂ℂ^N so that for each patch there is a function o on Ω with o_↑A○G = v_↑U which is strongly (weakly) q-convex of the same class.

Let Y be an analytic set in a complex space X and ψ a strongly q-convex C^∞ function on Y. For all continuous δ > 0 on Y, ∃ φ strongly q-convex C^∞ on a neighborhood V of Y with ψ ≤ φ_↑Y < ψ + δ. If Y is an analytic subvariety in a complex space X, with an almost PSH v on X with v = -∞ on Y with logarithmic poles, and v ∈ C^∞(X \ Y). If Y is a strongly q-complete analytic subset, then it has a fundamental family of strongly q-complete neighborhoods V in X. An open subset U of a complex X is q-Runge (q-Runge complete) in X if ∀ compact L ⊂ U ∃ smooth exhaustion ψ on X and a sublevel set X_b of ψ with L ⊂ X_b ⋐ U and ψ is strongly q-convex on X without the closure of X_b.