The Andreotti-Vesentini separation theorem and global homotopy representation

Abstract

Notation. We denote byC0 s,r(X,E) the space of continuousE-valued (s, r)forms on X (we omit E, when E is the trivial line bundle), by Z0 s,r(X,E) the subspace of ∂-closed forms, and byE0 s,r(X,E) the subspace of ∂-exact forms (E0 s,0(X,E) := {0}). As usual, H(X,E) : = Z0 s,r(X,E)/E 0 s,r(X,E). 0.1. Definition. X will be called q-concave-q∗-convex where q, q∗ are integers with 1 ≤ q ≤ n− 1 and 0 ≤ q∗ ≤ n− 1 if X is connected and there exists a real C2 function ρ on X such that if