On the theory of q-complete spaces

Let X be a complex space and f : X → I R a real valued function. Then f is said to be q-convex if for any x ∈ X there exist a neighborhood U which is biholomorphic to a closed analytic set in an open set Ω ⊂ I C and a function g ∈ C(Ω) such that i∂∂g has at most q − 1 zero or negative eigenvalues at each point of Ω and f |U = g|U . The space X is called q-complete if there exists an exhaustion function f on X such that f is q-convex on X. Let X be a q-complete space and D an open subset of X. Then D is said to be q-Runge in X if for any compact set K ⊂ D there is a q-convex exhaustion function f on X such that K ⊂ {x ∈ X : f(x) < 0} ⊂⊂ D. In a complex space X, an open set Ω is said to be locally q-complete if every