Complex Analytic and Differential Geometry 2025, 27 - q-Convexity Properties in Top Degrees

A strongly q-convex ψ on M, ε > 0, has a hermitian metric ω on M with increasing, indexed eigenvalues of the Hessian form id'd''ψ wrt. ω satisfy γ₁ ≥ -ε and γ_q = ... = γ_n = 1. Open sets U, W of M with all connected U_s having a connected component W_t(s) with W_t(s)∩U_s ≠ ∅, the same for W_t(s) without the closure of U_s. There is then a v ∈ C^∞(M, ℝ), v ≥ 0 with support contained in the union of closures of U, W so that v is strongly ω-subharmonic and positive definite on U. A connected, locally connected, locally compact topological space X with relatively compact open subset U, and U' the union of U with all connected components of X without U, then U' is open and relatively compact, and its complement has only finitely many connected components, all of which non-compact. Every n-dim connected non-compact complex manifold M has a strongly subharmonic exhaustion function wrt. hermitian metric ω. M is strongly n-complete. If X has maximal dim n, then X is always strongly (n+1)-complete. If X has no compact irreducible component of dim n, then X is strongly n-complete. If it has only finitely many irreducible components of dim n, then it's strongly n-convex. A connected non-compact n-dim complex manifold and U and open subset of M, then U is n-Runge complete in M iff M\U has no compact connected component.

A strongly q-complete manifold M, q≥1, and hoolomorphic vector bundle E over M, then H^k(M, O(E)) = 0 for k ≥ q and if U is a q-Runge complete open subset of M, and every d''-closed form h ∈ C^∞_{0, q-1}(U, E) can be approximated uniformly with its dervatives on compact subsets of U by a sequence of global d''-closed forms. For a coherent sheaf S of O_X modules on an analytic scheme (X, O_X), with quotient topology on S(U) independent from choice of exact sequence of the sheaf. If S is a coherent analytic sheaf over an analytic scheme (X, O_X), X compact, then all its cohomology groups are finite-dim and Hausdorff. An S-distinguished patch A ⊂ Ω of X with U' ⊂ U ⊂ A open, and U' q-Runge complete in U, then the restriction map H^q-1(U, S) → H^q-1(U', S) has a dense range. If (X, O_X) is a strongly q-convex analytic scheme, then H^k(X, S) Hausdorff and finite-dim for k ≥ q. If additionally U is q-Runge open in X, q ≥ 1, then the restriction map H^k(X, S) → H^k(U, S) is isomorphic for k ≥ q, and the restriction map H^q-1(X, S) → H^q-1(U, S) has dense range. If X is strongly q-complete, then H^k(X, S) = 0, k ≥ q. All sublevel sets X_c ⊃ K, H^k(X_c, S) is Hausdorff and finite-dim when k ≥ q. If also d > c, then the restriction map from X_d to X_c is isomorphic for k ≥ q and has dense range if k = q - 1.