Sheaf Cohomology 00: Roadmap
Sheaf cohomology / 00
Roadmap
This is the coverage audit and reading map for Part IV, "Sheaves and Cohomology on Complex Manifolds," in Complex Analysis and Riemann Surfaces.
Part IV of the PDF begins with concrete holomorphic line bundles and divisors, then builds the language of sheaves, Cech cohomology, derived-functor sheaf cohomology, exact sequences, fine resolutions, de Rham and Dolbeault comparisons, and finally the relation among divisors, degree, curvature, and the fundamental theorem of algebra. This series follows that development but supplies the background that is usually scattered across several sources.
Before the sheaf language: de Rham in one sentence
De Rham cohomology starts from the concrete quotient
\[H^k_{\mathrm{dR}}(X)=\frac{\text{closed }k\text{-forms}}{\text{exact }k\text{-forms}}.\]In positive degree, a closed form is locally a derivative, but it may not be a derivative globally; the obstruction is detected by periods over cycles. Article 10 begins with this quotient viewpoint before explaining why fine sheaves identify it with \(H^k(X,\underline{\mathbb R})\).
Coverage audit
Background supplied beyond the PDF
The notes expand the local-to-global mechanism behind sheaves, the Cech representative of a line bundle, the reason exactness is tested on stalks rather than only on global sections, the proof anatomy of long exact sequences, and the role of fine resolutions in computing cohomology. The differential-forms series is the natural prerequisite for Article 10; see Differential Forms 00. The divisor and canonical-bundle material also connects to Riemann-Roch 04 and Riemann-Roch 09.
Notation
Throughout, \(X\) denotes a complex manifold, and most geometric examples are compact Riemann surfaces. The structure sheaf is \(\mathcal O_X\), its sheaf of nowhere-zero holomorphic functions is \(\mathcal O_X^*\), the sheaf of smooth real or complex functions is \(\mathcal C^\infty_X\), and the sheaf of smooth \(k\)-forms is \(\mathcal A_X^k\). For a divisor \(D=\sum_p n_p p\), the associated sheaf is
\[\mathcal O_X(D)(U)= \{f\in \mathcal M_X(U): \operatorname{ord}_p(f)+n_p\ge 0 \text{ for every }p\in U\}.\]Cech cochains for an open cover \(\mathfrak U=\{U_i\}\) are denoted
\[C^q(\mathfrak U,\mathcal F)=\prod_{i_0<\cdots<i_q} \mathcal F(U_{i_0}\cap\cdots\cap U_{i_q}).\]The symbol \(H^q(X,\mathcal F)\) means derived-functor sheaf cohomology. For good covers and acyclic covers it agrees with the Cech cohomology computed from sufficiently refined covers.
Reading order
Articles 01-03 treat line bundles and divisors before general sheaves appear. Articles 04-07 build sheaf language and cohomology. Articles 08-10 explain how exact sequences and fine resolutions compute geometric invariants. Articles 11-12 return to compact Riemann surfaces, where Chern class, curvature, divisors, and degree meet.