Linear problems of complex analysis

This article attempts to give a linearized form of the basic theorems of complex analysis (the Oka–Cartan theory). With this aim we study simultaneously: a) the isomorphism problem for spaces of holomorphic functions $H(M)$ and $H(D^n)$, $n=\dim_{\mathbf C}M$; b) the existence of a linear separation of singularities for the space $H(U)$, where $U=U_0\cap U_1$, and $U_k$ ($k=0, 1$) are holomorphically convex domains in a complex manifold $M$, and, in a more general setting, the splitting of the Čech complex of a coherent sheaf over a holomorphically convex domain $V$; c) the existence of a linear extension for holomorphic functions on a submanifold $M\subset\Omega$, and more generally, the splitting of a global resolution of a coherent sheaf. In several cases (for strictly pseudoconvex domains) these questions can be answered affirmatively. The proofs are based on the theory of Hilbert scales and bounds for solutions of the $\bar\partial$-problem in weighted $L^2$-spaces. Counterexamples show that the same questions may also have negative answers.