Presentation and Čech quotient topologies for coherent analytic sections

Let $M$ be a finite-dimensional complex manifold, let ${\mathcal{O}}_M$ be its sheaf of holomorphic functions, and let $\mathcal{F}$ be a coherent ${\mathcal{O}}_M$-module sheaf. On an open set $V$ contained in a coordinate chart and in a neighborhood with a finite presentation ${\mathcal{O}}_V^{p_1}\to {\mathcal{O}}_V^{p_0}\stackrel{\pi }{\to }\mathcal{F}{\mathrm{\mid }}_V\to 0$, where $p_0,p_1$ are finite nonnegative ranks, require that the section map $\mathrm{\Gamma }(V,{\mathcal{O}}_V^{p_0})\to \mathrm{\Gamma }(V,\mathcal{F})$ is surjective. Give $\mathcal{O}(V{)}^{p_0}$ the compact-convergence topology generated by the seminorms $f\mapsto {\sup }_{z\in K}\parallel f(z)\parallel$ for compact $K\subset V$, and give $\mathrm{\Gamma }(V,\mathcal{F})$ the quotient topology induced by that surjection. On any open subset covered by countably many such sets, use the topology of the closed subspace of the product of their local section spaces consisting of compatible families. For a finite ordered open cover $\mathcal{U}=(U_i)$ whose nonempty finite intersections have these section topologies, give each Čech cochain space

$C^q(\mathcal{U},\mathcal{F}):={\prod }_{i_0<\cdots <i_q}\mathrm{\Gamma }(U_{i_0}\cap \cdots \cap U_{i_q},\mathcal{F}),(\delta c{)}_{i_0\cdots i_{q+1}}={\sum }_{\nu =0}^{q+1}(-1{)}^{\nu }{c}_{i_0\cdots \widehat{i_{\nu }}\cdots i_{q+1}}{\mid }_{U_{i_0}\cap \cdots \cap U_{i_{q+1}}}\mathrm{.}$

the finite-product topology, give its cocycle space the subspace topology, and give the quotient of cocycles by coboundaries the Čech quotient topology. The accompanying Fréchet-quotient lemma proves local chart and presentation independence; the compact projective-space finiteness theorem proves independence of its admissible finite acyclic cover topology.

Coherent module sheaves on complex manifolds

Fréchet spaces used in analytic coherent cohomology

Jean-Pierre Demailly, Complex Analytic and Differential Geometry, Chapter IX §4.C

Henri Cartan, Un théorème de finitude, Exposé 17