Donaldson — Riemann Surfaces (Fast Track 1.07) — Audit + Gap Plan

Book: Simon Donaldson, Riemann Surfaces (Oxford Graduate Texts in Mathematics 22, Oxford University Press, 2011). ISBN 978-0-19-960674-0.

Fast Track entry: 1.07 (Riemann-surfaces / one-complex-variable slot of the geometric-analysis strand; Donaldson paired with Forster, Miranda, Farkas-Kra as the differential-geometric / bundle-theoretic anchor).

Purpose: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite per orchestration protocol). Punch-list of new units + deepenings to reach the equivalence threshold (docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4). Not a full P1 audit (no line-number-level exercise inventory).

The audit surface is comparable to Cannas: the Codex Riemann-surfaces chapter already ships 21 units; Donaldson is named in tier_anchors on 06.03.03-uniformization-theorem (Ch. 6) and 06.05.02-holomorphic-line-bundle (Ch. 8). Topic-level coverage is high; real gaps are at depth — sheaf cohomology (Čech), Serre duality on curves, Riemann-Roch proof, Gauss-Manin, PDE / Dirichlet-energy uniformisation, theta-function machinery beyond definition, Jacobi inversion, Schottky, Hodge decomposition.

§1 What Donaldson's book is for

Donaldson is the canonical differential-geometric and bundle-theoretic introduction to Riemann surfaces. Where Forster (1.07-paired) anchors the sheaf-theoretic and complex-analytic perspective — Čech cohomology, Mittag-Leffler, finiteness via Schwartz's theorem — and Miranda anchors the algebraic-curves perspective — divisors, linear systems, projective embeddings — Donaldson sits where the differential geometer enters the subject. The book is organised as 14 chapters across three Parts:

Part I — Preliminaries (Ch. 1-2): basic complex analysis on the plane (Cauchy, Liouville, Riemann mapping for the disc), surfaces and Riemann surfaces from the differentiable viewpoint (charts, atlases, the standard examples — Riemann sphere, tori, hyperelliptic surfaces, plane curves).
Part II — Basic Theory (Ch. 3-7): differential forms on Riemann surfaces (Hodge star, harmonic forms, $\overset{ˉ}{\partial}$ operator), the main theorem (existence of meromorphic functions on compact Riemann surfaces via the Hodge / Dirichlet-energy method — Donaldson's signature treatment), proofs of the main theorem (Hilbert space methods, weak solutions, regularity), the Riemann-Roch theorem (with full proof using Serre duality on curves), uniformisation (in the genus 0 / 1 / ≥ 2 cases via PDE / harmonic function methods).
Part III — Deeper Theory (Ch. 8-14): holomorphic line bundles and divisors as the bundle-theoretic translation of Riemann-Roch; further topics including embeddings into projective space (Kodaira-type criteria for curves), Picard group structure, theta functions and the Jacobi inversion problem (proving $g =$ dimension of theta divisor and Jacobi's theorem on surjectivity of the Abel-Jacobi map), Riemann's bilinear relations, Schottky's theorem / Schottky problem, the Gauss-Manin connection (variation of periods over moduli of curves), and a final chapter touching on moduli of curves / Teichmüller theory.

Distinctive Donaldson choices:

Differential-geometric throughout. Donaldson is a Fields-medallist differential geometer (gauge theory, four-manifold invariants). The book's vocabulary is Hodge star, $\overset{ˉ}{\partial}$ as Cauchy-Riemann operator, harmonic 1-forms, Dirichlet energy, Hodge decomposition. Where Forster develops everything via sheaves and Mittag-Leffler, Donaldson develops everything via PDE on a Riemannian surface with compatible complex structure. Codex's existing chapter is sheaf-/divisor-theoretic in the Forster tradition; the load-bearing PDE / harmonic theory is not present.
Sheaf cohomology of holomorphic line bundles via Čech. Ch. 8-9 develop $H^{q} (M, O (L))$ — the algebraic vehicle for Riemann-Roch. Codex has the line-bundle unit but no genuine Čech-cohomology unit; the sheaf-cohomology layer is missing.
Serre duality on curves, proved. Ch. 9: $H^1(M, L) \cong H^0(M, K \otimes L^{-1})^*$ via Hodge theory from Ch. 5-7. Codex's Riemann-Roch unit treats Serre duality as a black box; no dedicated Serre-duality unit exists.
Riemann-Roch with full proof. Ch. 9 proves $\dim H^0(L) - \dim H^1(L) = \deg L + 1 - g$ using sheaf cohomology + Serre duality + the long exact sequence for $\mathcal{O}(L) \to \mathcal{O}(L+P) \to \mathbb{C}_P$. Codex's RR unit states the result with intuition but does not carry out the inductive divisor-bumping proof.
Uniformisation via PDE. Ch. 7 proves the universal cover is $P^{1}$ , $C$ , or $D$ via Dirichlet-energy / Green's-function PDE machinery. Codex has the unit (Donaldson cited in tier_anchors) but the proof is sketchy / classical-style rather than Donaldson's PDE proof.
Theta functions with Jacobi inversion. Ch. 11 proves Jacobi's inversion theorem (Abel-Jacobi $Sym^{g} (M) \to J (M)$ surjective) and Riemann's vanishing theorem. Codex has the theta-function and Abel-Jacobi units but the inversion bridge between them is missing.
Riemann's bilinear relations and Schottky problem. Ch. 11-12. Period matrix in the Siegel upper half space; Schottky's $g = 4$ theorem as apex. Codex has period-matrix unit; bilinear relations only mentioned, Schottky absent.
Gauss-Manin connection. Ch. 12-13. Variation of Hodge structure as complex structure varies in moduli; flat connection on the cohomology bundle. Codex has zero coverage — largest topic-level gap on Donaldson's side.
Hodge decomposition for curves. Ch. 5 / Ch. 11. $H^1(M, \mathbb{C}) = H^{1,0} \oplus H^{0,1}$ via harmonic 1-form theory. Structural input for period matrix and bilinear relations. Codex's holomorphic-1-form unit has the definition but does not carry out the Hodge decomposition explicitly.
Modern algebraic-curves outlook. 2011 textbook with awareness of the projective-embedding (Kodaira-type) criterion, Picard group structure, divisor class group. Forster lacks this.
Examples-driven from Ch. 2. Riemann sphere, $C /Λ$ with Weierstrass $℘$ , hyperelliptic $y^{2} = p (x)$ , smooth plane curves — running examples throughout.
Concise — 286 pages. Production estimates below reflect that Donaldson has fewer micro-topics but proves headline theorems in greater PDE/differential-geometric depth.

Donaldson ends before higher-genus moduli theory (no Teichmüller in depth, no Deligne-Mumford, no Mumford GIT — points to Mumford / Hubbard).

§2 Coverage table (Codex vs Donaldson)

Cross-referenced against the 21-unit Codex Riemann-surfaces chapter (find content/06-riemann-surfaces -name "*.md" | sort). ✓ = covered at Donaldson-equivalent depth, △ = topic present but Codex unit shallower (typically templated v0.5 Strand-B prose), ✗ = not covered.

Part I — Preliminaries (Ch. 1-2)

Donaldson topic	Codex unit	Status	Note
Holomorphic functions on $C$ , Cauchy theorem, Liouville	`06.01.01-holomorphic-function`; `06.01.02-cauchy-integral-formula`	✓	Both shipped at the right depth; Donaldson's Ch. 1 is review.
Meromorphic functions, residue theorem	`06.01.05-meromorphic-function`; `06.01.03-residue-theorem`	✓	Both shipped.
Analytic continuation	`06.01.04-analytic-continuation`	✓	Shipped.
Riemann mapping theorem (the disc case)	`06.01.06-riemann-mapping-theorem`	✓	Shipped; Donaldson Ch. 1 covers same depth.
Riemann surface (chart-atlas definition)	`06.03.01-riemann-surface`	✓	Shipped. Donaldson cited.
Genus, topological classification of compact surfaces	`06.03.02-genus-riemann-surface`	✓	Shipped. Donaldson cited.
Standard examples — Riemann sphere, complex torus $C /Λ$ , hyperelliptic surface, smooth plane curves	—	△	The general examples appear scattered in Master sections of the existing units, but no dedicated "running examples of Riemann surfaces" unit. Donaldson Ch. 2 is essentially this.
Branch points and ramification	`06.02.01-branch-point-ramification`	✓	Shipped.

Part II — Basic Theory (Ch. 3-7)

This is where Donaldson's distinctive PDE / differential-geometric approach lives. Codex coverage thins here.

Donaldson topic	Codex unit	Status	Note
Differential forms on a Riemann surface; Hodge star $⋆$ ; $d, \overset{ˉ}{\partial}, \partial$ operators	`06.06.01-holomorphic-one-form` (partial)	△	The holomorphic-1-form unit defines $ω \in Ω^{1, 0}$ but does not develop the full $(p, q)$ -form / Hodge-star machinery. Donaldson Ch. 3 gives this as the working vocabulary for everything that follows. Foundational depth gap.
Harmonic 1-forms, Hodge decomposition $H^{1} (M, C) = H^{1, 0} \oplus H^{0, 1}$	partial in `06.06.01`	△	Holomorphic-1-form unit mentions $dim H^{0} (M, Ω^{1}) = g$ but does not carry out the Hodge decomposition. Donaldson Ch. 5; depth gap, high priority.
Main theorem (existence of meromorphic functions on compact $M$ via Hodge / Dirichlet-energy method)	—	✗	Gap (high priority). Donaldson Ch. 4-5. The core PDE result the book is structured around: prove that on any compact Riemann surface there exist non-constant meromorphic functions, by solving the Dirichlet problem for harmonic functions and constructing the meromorphic function from a harmonic primitive.
Hilbert-space PDE machinery: weak solutions to $\overset{ˉ}{\partial} u = f$ , Sobolev / regularity, Friedrichs extension	—	✗	Gap. Donaldson Ch. 5-6. Setup for the main theorem.
Sheaf cohomology of holomorphic line bundles $H^{q} (M, O (L))$ via Čech cocycles	—	✗	Gap (high priority). Donaldson Ch. 8. The full Čech construction — open covers, Čech cochains, cocycles, coboundaries, $H^{0}$ as global sections, $H^{1}$ as obstruction — is not present anywhere in `06.`. Foundational for Riemann-Roch.*
Serre duality on curves: $H^{1} (M, L) ≅ H^{0} (M, K \otimes L^{- 1})^{*}$	—	✗	Gap (high priority). Donaldson Ch. 9. The duality pairing realised via the residue / cup product. Direct input to Riemann-Roch proof.
Riemann-Roch theorem with full proof	`06.04.01-riemann-roch-compact-riemann-surfaces`	△	Statement and intuition shipped; Donaldson cited in Master tier. The actual divisor-bumping induction proof using Serre duality is not carried out — current Master section gives the consequence and worked degree-counting, not the proof. Depth gap, high priority.
Uniformisation theorem (universal cover is $P^{1}$ , $C$ , or $D$ ) — full PDE proof	`06.03.03-uniformization-theorem`	△	Statement shipped; Donaldson cited in `tier_anchors` (Donaldson Ch. 6). The Master section gives a sketch that mentions monodromy / covering theory but does not carry out Donaldson's PDE proof (construct Green's function via Dirichlet energy, integrate to uniformising map). Depth gap, high priority.

Part III — Deeper Theory (Ch. 8-14)

Donaldson topic	Codex unit	Status	Note
Holomorphic line bundles on a Riemann surface, transition functions	`06.05.02-holomorphic-line-bundle`	△	Shipped, Donaldson cited (Donaldson Ch. 8). The cocycle definition is present; the bundle-theoretic translation of Riemann-Roch ( $dim H^{0} (L) - dim H^{1} (L) = de g L + 1 - g$ as a statement about line bundles, equivalent to the divisor form) is partly there. Depth assessment: roughly Donaldson-equivalent at definition level, shallower at Riemann-Roch-translation level.
Divisor on a Riemann surface, principal divisor, divisor class group, $Pic (M)$	`06.05.01-divisor-riemann-surface`	△	Divisor unit shipped; the equivalence " $Pic (M) = Cl (M) = Div (M) / PrinDiv (M)$ " is stated but the structure of $Pic (M)$ as $Pic^{0} (M) \oplus Z$ (via degree) and the identification $Pic^{0} (M) ≅ J (M)$ is not developed cleanly. Donaldson Ch. 10 has this.
Riemann-Hurwitz formula	`06.05.03-riemann-hurwitz-formula`	✓	Shipped; Donaldson Ch. 4 covers same depth.
Holomorphic 1-forms, $dim H^{0} (Ω^{1}) = g$	`06.06.01-holomorphic-one-form`	△	Statement shipped; the proof via Hodge decomposition is not carried out (depth gap above).
Period matrix $Π \in M_{g \times 2 g} (C)$ , $A$ / $B$ -periods	`06.06.02-period-matrix`	△	Shipped; the Riemann bilinear relations (period matrix is in Siegel upper half space $H_{g}$ ) are not proved. Donaldson Ch. 11. Depth gap.
Jacobian variety $J (M) = C^{g} /Λ$ where $Λ$ is the period lattice	`06.06.03-jacobian-variety`	△	Shipped; the principally-polarised-abelian-variety structure (via theta function) is mentioned but not developed. Donaldson Ch. 11.
Abel-Jacobi map $A : Div^{0} (M) \to J (M)$ , Abel's theorem (kernel = principal divisors)	`06.06.04-abel-jacobi-map`	△	Abel's theorem shipped.
Jacobi inversion theorem (Abel-Jacobi $Sym^{g} (M) \to J (M)$ surjective)	—	✗	Gap (high priority). Donaldson Ch. 11. The theorem that completes Abel — together they say $Sym^{g} (M) \to J (M)$ is birational.
Theta function on $J (M)$	`06.06.05-theta-function`	△	Shipped; basic definition $θ (z, τ) = \sum_{n \in Z^{g}} exp (π i n^{T} τ n + 2 π i n^{T} z)$ present. Riemann's vanishing theorem (zeros of theta = Abel-Jacobi image of effective divisors of degree $g - 1$ shifted by Riemann constant) absent. Donaldson Ch. 11.
Riemann's bilinear relations	—	✗	Gap. Donaldson Ch. 11. The integrality / symmetry / positivity conditions on the period matrix that characterise Jacobians among complex tori.
Schottky's theorem / Schottky problem	—	✗	Gap (priority-2/3). Donaldson Ch. 12. Classifies Jacobians among principally polarised abelian varieties. The Schottky problem in $g \geq 4$ .
Gauss-Manin connection (variation of periods over moduli)	—	✗	Gap (priority-2, large topic). Donaldson Ch. 12-13. Variation of Hodge structure for the universal family of Riemann surfaces; the flat connection encoding period variation. Largest single Donaldson-distinctive topic-level gap.
Embedding of curves in projective space; very-ample line bundles; canonical embedding for $g \geq 2$ non-hyperelliptic	—	✗	Gap. Donaldson Ch. 10. Codex's projective-geometry coverage in `04.*` is general; the curves-specific Kodaira-style criterion is not present.
Moduli of Riemann surfaces / Teichmüller theory pointers	—	✗	Pointer-level only in Donaldson; defer.

Several-variables (Ch. 7 several-variables — Codex's existing chapter section)

Codex has two units in 07-several-variables/ (06.07.01-holomorphic-several-variables, 06.07.02-hartogs-phenomenon) which are not part of Donaldson's narrative — Donaldson is strictly one-variable. These units are anchored from Hörmander / Krantz, not Donaldson, and require no Donaldson-equivalence work.

Aggregate coverage estimate

Theorem layer: ~70% topic-level, ~35% Donaldson-equivalent proof-depth. Gap concentrated in (a) PDE / Hodge / Čech infrastructure (Ch. 5-9), (b) Riemann-Roch proof (unit exists, no proof), (c) Jacobian-side deepening (bilinear relations, Jacobi inversion, Riemann vanishing, Schottky), (d) Gauss-Manin / VHS block (absent). After priority-1: topic ~92%, proof-depth ~70%. After priority-1+2: ~93% proof-depth.
Exercise layer: Donaldson's ~80 exercises vs. Codex's templated 7-block. Defer to dedicated exercise-pack pass.
Worked-example layer: ~40%. Donaldson runs four standard examples (Riemann sphere, torus, hyperelliptic, plane curves) systematically; Codex uses them as one-off references.
Notation layer: ~80% aligned. Donaldson uses $M$ (Codex mixes $M / X /Σ$ — chapter-wide cleanup issue, not Donaldson-specific), $K$ , $g$ , $O (L)$ , $J (M)$ , $Π$ , $θ$ . No notation/donaldson.md needed.
Sequencing layer: ~75%. Codex DAG follows Forster flow; needs new edges harmonic 1-forms → Hodge decomp → Serre duality → Riemann-Roch.
Intuition layer: ~50%. Sheaf-/divisor-theoretic intuition decent; differential-geometric / PDE intuition essentially absent.
Application layer: ~60%. $℘$ from theta, low-genus periods partial; Schottky / moduli applications absent.

§3 Gap punch-list (P3-lite — units to write or deepen, priority-ordered)

The Codex Riemann-surfaces chapter is mature in topic coverage. Most of the work below is deepening existing units to Donaldson-proof-depth, plus a small number of genuinely new units (Čech cohomology, Serre duality, Hodge decomposition, Jacobi inversion, Gauss-Manin). The recommended slot range is 06.04.* for cohomology / Riemann-Roch infrastructure, 06.06.* for Jacobian-side deepenings and new units, and a new chapter 06.08-moduli-and-variation/ for the Gauss-Manin / moduli block.

Priority 1 — load-bearing PDE / cohomology infrastructure and signature theorems

These items either provide infrastructure used silently elsewhere (Čech cohomology, harmonic / Hodge decomposition, Serre duality) or are Donaldson's signature proofs of named theorems (Riemann-Roch, uniformisation via PDE). Without them the Riemann-surfaces chapter cannot honestly claim Donaldson-equivalence.

06.04.02 Čech cohomology of holomorphic line bundles. Donaldson Ch. 8 anchor; Cartan-Eilenberg 1956 / Godement 1958 as originator-citations for sheaf-Čech machinery, Serre 1955 Faisceaux algébriques cohérents for the holomorphic-line-bundle case. Three-tier; ~2200 words. Master section: open covers ${U_{α}}$ , Čech $q$ -cochains $C^{q}$ on intersections, coboundary $δ$ , $H^{q} (M, O (L))$ for line bundle $L$ ; $H^{0}$ as global sections, $H^{1}$ as obstruction class; refinement-independence; short exact sequence of sheaves $\to$ long exact sequence of Čech cohomology. Worked examples: $H^0(\mathbb{P}^1, \mathcal{O}(d)) = \mathbb{C}^{d+1} $;$ H^1(\mathbb{P}^1, \mathcal{O}(d)) = 0$ for $d \geq - 1$ . Highest priority — every other priority-1 item builds on it.
06.04.03 Hodge decomposition for compact Riemann surfaces. Donaldson Ch. 5 anchor; Hodge 1941 Theory and Applications of Harmonic Integrals originator-citation; Weyl 1913 Die Idee der Riemannschen Fläche for the Riemann-surface case (Dirichlet's principle). Three-tier; ~2000 words. Master section: $\Omega^(M) = \bigoplus_{p+q=} \Omega^{p,q}(M)$; harmonic 1-forms $H^{1} (M) = ker (Δ)$ ; Hodge theorem $H^1(M, \mathbb{C}) = \mathcal{H}^{1,0} \oplus \mathcal{H}^{0,1}$; for compact Riemann surfaces, $dim H^{1, 0} = dim H^{0, 1} = g$ (the genus). Master section gives the Hilbert-space proof: harmonic 1-form is unique representative of cohomology class minimising $L^{2}$ -norm. Used by items 3-6.
06.04.04 Serre duality on curves. Donaldson Ch. 9 anchor; Serre 1955 Un théorème de dualité originator-citation. Three-tier; ~2000 words. Master section: for a holomorphic line bundle $L$ on compact Riemann surface $M$ of genus $g$ , the residue / cup product pairing $H^1(M, L) \otimes H^0(M, K \otimes L^{-1}) \to H^1(M, K) = \mathbb{C} $i s n o n - d e g e n er a t e; h e n ce$ H^1(M, L) \cong H^0(M, K \otimes L^{-1})^*$. Master section sketches the harmonic-theory proof via the Hodge decomposition of item 2. Worked example: Riemann sphere, $L = O (d)$ , both sides explicitly computable.
Deepen 06.04.01-riemann-roch-compact-riemann-surfaces (full proof). Replace the templated / consequence-style "Key theorem with proof" with the actual divisor-bumping induction: short exact sequence $0 \to \mathcal{O}(L) \to \mathcal{O}(L+P) \to \mathbb{C}_P \to 0 $f or a p o in t$ P $, in d u ce d l o n g e x a c t se q u e n ce in$ H^$, both sides of Riemann-Roch shift by 1, base case $L = O$ gives $1 - g = 0 + 1 - g$ . Citing item 3 (Serre duality) for the $H^1 = H^0(K-L)^$ identification. Donaldson Ch. 9. No new unit ID; rewrite of the Intermediate "Key theorem" and Master "Full proof" sections.
Deepen 06.03.03-uniformization-theorem (PDE proof). Replace the current covering-theoretic sketch with Donaldson's PDE proof: construct the Green's function on $M$ via the Dirichlet-energy minimisation, exponentiate to a holomorphic function with prescribed pole, use the holomorphic function to construct the uniformising map to $P^{1}$ / $C$ / $D$ depending on genus. Donaldson Ch. 6. No new unit ID; rewrite of the Master "Full proof" section. The classical-style sketch can remain as alternative-proof commentary; the PDE proof becomes the primary exposition. Deepening, high priority — the unit names Donaldson in tier_anchors so the proof should match Donaldson.
06.06.06 Jacobi inversion theorem. Donaldson Ch. 11 anchor; Jacobi 1832-1834 (originator) joint with Riemann's 1857 Theorie der Abel'schen Functionen for the modern formulation. Three-tier; ~1800 words. Master section: the symmetric product $Sym^{g} (M)$ is mapped to $J (M)$ by Abel-Jacobi; the theorem says this map is surjective and birational; combined with Abel's theorem (item 06.06.04), $Sym^{g} (M) / \sim≅ J (M)$ . Proof via the theta function (item 7 below) — vanishing locus computation (Riemann's vanishing theorem). Worked example: genus 1 — Abel-Jacobi is a bijection $M \to J (M)$ (the elliptic-curve identification with its Jacobian).
Deepen 06.06.05-theta-function (Riemann's vanishing theorem). Replace the templated proof / definition-only treatment with the substantive content of Donaldson Ch. 11: the theta function $\theta(z, \tau) $i s ah o l o m or p hi csec t i o n o f a s p ec i f i c l in e b u n d l eo n$ J(M)$, its zero locus is the theta divisor $Θ \subset J (M)$ , and $Θ$ is the image of $Sym^{g - 1} (M)$ under Abel-Jacobi shifted by the Riemann constant $κ$ . This is Riemann's vanishing theorem. No new unit ID; rewrite of the Intermediate "Key theorem" and Master "Full proof" sections.

Priority 2 — Donaldson-distinctive depth gaps and the Gauss-Manin block

These items either replace templated content with Donaldson-anchored prose on existing units, introduce the Riemann-bilinear-relations and Schottky-theorem block that Donaldson treats as substantive, or open the moduli / variation-of-Hodge-structure block (the largest Donaldson-distinctive topic-level gap).

06.06.07 Riemann's bilinear relations. Donaldson Ch. 11 anchor; Riemann 1857 originator. Three-tier; ~1800 words. Master section: choose canonical homology basis $A_1, \dots, A_g, B_1, \dots, B_g $w i t h$ A_i \cdot B_j = \delta_{ij}$; for holomorphic 1-forms $ω_{1}, \dots, ω_{g}$ normalised by $\int_{A_i} \omega_j = \delta_{ij} $, t h e p er i o d ma t r i x$ \Pi_{ij} = \int_{B_i} \omega_j $s a t i s f i es (i)$ \Pi^T = \Pi $(sy mm e t r y), (ii)$ \mathrm{Im}, \Pi > 0 $(p os i t i v i t y); h e n ce$ \Pi \in \mathfrak{H}_g$, the Siegel upper half space. Bridge to Schottky problem (item 9).
06.06.08 Schottky's theorem and the Schottky problem. Donaldson Ch. 12 anchor; Schottky 1888 / Schottky-Jung 1909 originator-citations for the genus-4 case; Mumford 1975 Tata Lectures on Theta II and Igusa for modern reformulations. Three-tier; ~1800 words. Master section: Schottky's classical $g = 4$ theorem (a quartic relation among period values), and the Schottky problem in general (which $H_{g}$ -points come from curves; the Jacobi locus inside the moduli of principally polarised abelian varieties).
06.08.01 Gauss-Manin connection. Donaldson Ch. 12-13 anchor; Manin 1958 / Katz-Oda 1968 originator-citations. Three-tier; ~2400 words. Largest single new unit in this plan. Master section: family of Riemann surfaces $π : M \to S$ over a base $S$ (parameter space, e.g. $H_{g}$ or moduli space $M_{g}$ ); the cohomology bundle $R^{1} π_{*} C$ is locally constant (topological invariance), so carries a flat connection — the Gauss-Manin connection; the Hodge bundles $R^{1} π_{*} O$ and $π_{*} Ω^{1}$ vary holomorphically; period matrix as multivalued function on $S$ . Worked example: family of elliptic curves $y^{2} = x (x - 1) (x - λ)$ over $\lambda \in \mathbb{C} \setminus {0, 1}$ — periods are hypergeometric functions of $λ$ ; the Gauss-Manin connection has regular singular points at $λ = 0, 1, \infty$ . Priority because largest topic-level gap, and bridges Riemann surfaces to algebraic geometry / Hodge theory more broadly.
06.08.02 Variation of Hodge structure on the Jacobian. Donaldson Ch. 13 anchor; Griffiths 1968-70 Periods of integrals on algebraic manifolds originator-citation. Three-tier; ~1800 words. Master section: the Hodge filtration $F^{1} \subset H^{1}$ varies holomorphically, $F^{1}$ has rank $g$ , transversality condition $\nabla F^{p} \subset F^{p - 1} \otimes Ω_{S}^{1}$ (Griffiths transversality). Bridge to Hodge theory in the algebraic geometry chapter (04.*).
Deepen 06.05.02-holomorphic-line-bundle (full RR translation). Add the bundle-theoretic restatement of Riemann-Roch: $\dim H^0(L) - \dim H^1(L) = \deg L + 1 - g$; the equivalence with the divisor form via $L \leftrightarrow [D]$ . Add the degree invariant $de g L = \int_{M} c_{1} (L)$ as the Chern-class integration. Donaldson Ch. 8-10. No new unit ID.
Deepen 06.05.01-divisor-riemann-surface (Picard group structure). Add the structural result $\mathrm{Pic}(M) = \mathrm{Pic}^0(M) \oplus \mathbb{Z}$ via the degree map; the identification $Pic^{0} (M) ≅ J (M)$ as complex Lie groups; the short exact sequence $0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^* \to 0$ and the connecting map giving the Chern class. Donaldson Ch. 10. No new unit ID.
06.04.05 Hilbert-space PDE for $\overset{ˉ}{\partial}$ . Donaldson Ch. 5-6 anchor; Hörmander 1965 originator. Three-tier; ~1500 words. Master section: weak solutions to $\overset{ˉ}{\partial} u = f$ for $f \in L^2 \Omega^{0,1} $; S o b o l e v s p a ce ma c hin er y;$ L^2$-projection onto harmonic forms; regularity (smooth $f$ ⇒ smooth $u$ ). Setup for items 2, 5; produces the main-theorem PDE proof Donaldson hangs his book on.
06.04.06 Existence of meromorphic functions on compact Riemann surfaces (the main theorem). Donaldson Ch. 4 anchor; Riemann 1857 originator. Three-tier; ~1700 words. Master section: starting from the harmonic-1-form / Dirichlet-energy machinery (items 2 and 14), construct a meromorphic function with prescribed pole at a given point on any compact $M$ ; consequence: every compact $M$ embeds projectively. Conceptual climax of Donaldson Part II.
Deepen 06.06.02-period-matrix (bilinear relations stated and proved). Cross-reference to item 8; flag $Π \in H_{g}$ . No new unit ID; addition to existing Master section.
Deepen 06.06.01-holomorphic-one-form (Hodge-decomposition proof of $dim H^{0} (Ω^{1}) = g$ ). Replace templated proof with harmonic-theory argument (cite item 2). No new unit ID.

Priority 3 — Donaldson-distinctive optional units (depth-completing)

These items round out Donaldson's content but are not strictly required for FT equivalence: items 18-19 are showcase examples that make the running-example layer match Donaldson; item 20 is the curves-projective embedding bridge; item 21 is moduli-space pointer.

06.03.04 Standard examples of Riemann surfaces (running examples). Donaldson Ch. 2 anchor. Three-tier; ~2000 words. Master section: detailed treatment of (a) Riemann sphere $P^{1} = C \cup {\infty}$ with the two-chart atlas; (b) complex torus $C /Λ$ with the Weierstrass $℘$ -function generating the field of meromorphic functions; (c) hyperelliptic surface $y^{2} = p (x)$ with $p$ degree $2 g + 1$ or $2 g + 2$ ; (d) smooth plane curves $F (x, y) = 0$ . Worked examples referenced from later units. Closes the worked-example-layer gap described in §2.
06.05.04 Embedding of curves in projective space; canonical embedding. Donaldson Ch. 10 anchor. Three-tier; ~1700 words. Master section: a line bundle $L$ on $M$ is very ample if it embeds $M \to P (H^{0} (L)^{*})$ ; Kodaira-style criterion for curves; the canonical embedding for $g \geq 2$ non-hyperelliptic via $K$ . Worked example: genus-3 curve embeds via $K$ as a quartic in $P^{2}$ ; genus-4 curve embeds via $K$ as the intersection of a quadric and a cubic in $P^{3}$ .
06.06.09 Principally polarised abelian variety structure on $J (M)$ . Donaldson Ch. 11 (footnote) anchor. Three-tier; ~1500 words. The polarisation $Θ \subset J (M)$ is principal; bridge to abelian varieties in algebraic geometry. Optional.
06.08.03 Survey: moduli of Riemann surfaces. Donaldson Ch. 13 (pointer) anchor; Mumford 1965 / Deligne-Mumford 1969 originator-citations. Master-only; ~900 words. Pointer to Hubbard / Farb-Margalit Teichmüller theory, Mumford GIT construction of $M_{g}$ , Deligne-Mumford compactification. Deferred unless Codex expands into moduli theory.

Priority 4 — Survey / pointer items, optional

06.04.07 Survey: sheaf cohomology in algebraic geometry. Master-only; ~700 words. Pointer to Hartshorne / Griffiths-Harris for the algebraic-geometry-of-curves treatment of the same cohomology machinery. Cross-link rather than expansion.
Notation crosswalk (notation/donaldson.md). Optional given the mostly-aligned conventions. Worth producing if a chapter-wide $M / X /Σ$ notation pass is undertaken; otherwise defer.

§4 Implementation sketch (P3 → P4)

Minimum Donaldson-equivalence batch = priority 1 only (items 1-7): 4 new units (06.04.02, 06.04.03, 06.04.04, 06.06.06) plus 3 deepenings (Riemann-Roch, uniformisation, theta function). Realistic production estimate (mirroring earlier Cannas / Arnold / Lawson-Michelsohn batches):

~3 hours per typical new unit (research + draft + validate at 27/27 + Lean stub + Bridge / Synthesis prose).
~5 hours for Čech cohomology unit (item 1) — large and load-bearing.
~1.5 hours per deepening.
Priority 1 totals: 1 large × 5 h + 3 typical × 3 h + 3 deepenings × 1.5 h = ~18.5 hours.
Priority 1+2 totals: 1 large × 5 h + 6 new typical × 3 h + 5 deepenings × 1.5 h = ~30.5 hours.

At 4-6 production agents in parallel, priority-1 fits in a 1-2 day window with one integration agent stitching outputs.

Batch structure.

Batch A (PDE / cohomology infrastructure, items 1, 2, 14, 3, ~14 h): opens new sections in 06.04-*. Produce together; load-bearing for everything downstream.
Batch B (Theorem proofs, items 4, 5, 15, ~9 h): depends on Batch A. Riemann-Roch proof (item 4) needs Serre duality (item 3); uniformisation (item 5) needs Hodge decomposition (item 2); main theorem (item 15) needs item 14.
Batch C (Jacobian-side deepening, items 6, 7, 8, 9, 16, 17, ~13 h): opens / extends 06.06-*. Largely independent of Batch B — only depends on Batch A (item 2 for Hodge decomposition feeding the period-matrix bilinear relations).
Batch D (Gauss-Manin block, items 10, 11, ~8 h): opens new 06.08-moduli-and-variation/ chapter. Largest priority-2 item. Independent of A-C, can run in parallel.
Optional Batch E (priority-3+4, items 18-23, ~10 h): after priority-1+2 lands.

Originator-prose targets (each priority-1 unit's Master section cites originator + Donaldson):

Čech cohomology (item 1): Cartan-Eilenberg 1956; Godement 1958 Théorie des faisceaux; Serre 1955 Faisceaux algébriques cohérents (Ann. Math. 61) for the holomorphic case.
Hodge decomposition (2): Hodge 1941; Weyl 1913 Die Idee der Riemannschen Fläche for the Riemann-surface case.
Serre duality (3): Serre 1955 Un théorème de dualité (Comm. Math. Helv. 29). Originator-prose mandatory.
Riemann-Roch (4 deepening): Riemann 1857; Roch 1865; Hirzebruch 1956 / Grothendieck 1957 (optional modern cite).
Uniformisation PDE (5 deepening): Klein 1882; Poincaré 1907 / Koebe 1907; Donaldson Ch. 6.
Jacobi inversion (6): Jacobi 1832-1834; Riemann 1857.
Riemann's vanishing (7 deepening): Riemann 1857; Mumford Tata Lectures on Theta I-III.
Bilinear relations (8): Riemann 1857. Originator voice mandatory.
Schottky (9): Schottky 1888 (J. reine angew. Math. 102).
Gauss-Manin (10): Manin 1958; Katz-Oda 1968 (J. Math. Kyoto 8).
VHS (11): Griffiths 1968-70 Periods of integrals on algebraic manifolds I-III.
Hilbert-space $\overset{ˉ}{\partial}$ (14): Hörmander 1965 (Acta Math. 113).

Notation crosswalk. Mostly aligned (per §2). No notation/donaldson.md needed; existing tier_anchors citing Donaldson Ch. N is sufficient. Pin $M$ as the Riemann-surface symbol for new Donaldson-anchored units (Codex currently mixes $M / X / \Sigma$ — chapter-wide cleanup is a separate maintenance pass).

DAG edges to add. New prerequisites for the priority-1+2 batch:

06.04.02 (Čech cohomology) ← {06.05.02-holomorphic-line-bundle, 06.04.03 Hodge}
06.04.03 (Hodge decomp) ← {06.06.01-holomorphic-one-form, 06.04.05 Hilbert-PDE}
06.04.04 (Serre duality) ← 06.04.03 Hodge; 06.04.04 → 06.04.01 Riemann-Roch
06.04.05 (Hilbert PDE for $\overset{ˉ}{\partial}$ ) ← {06.01.01, 06.06.01-holomorphic-one-form}
06.04.06 (main theorem / existence of meromorphic functions) ← {06.04.05, 06.04.03}; → 06.05.04 curves-embedding
06.06.06 (Jacobi inversion) ← {06.06.04-abel-jacobi-map, 06.06.05-theta-function}
06.06.07 (Riemann bilinear relations) ← {06.06.02-period-matrix, 06.04.03 Hodge}
06.06.08 (Schottky) ← {06.06.07, 06.06.06}
06.08.01 (Gauss-Manin) ← {06.06.02-period-matrix, 06.04.03, 06.06.03-jacobian-variety}
06.08.02 (VHS) ← 06.08.01
06.05.04 (curves embedding) ← {06.05.02, 06.04.06}

Composite Donaldson + Forster batch recommendation. Forster's book shares the algebraic-machinery side (sheaves, divisors, Riemann-Roch statement) with Donaldson. The Čech cohomology unit (item 1) and the Riemann-Roch deepening (item 4) cover both books with one production pass. Defer the Forster audit to the same campaign window so that these heavy units are produced once and serve both equivalences.

§5 What this plan does NOT cover

Line-number-level inventory of every theorem / exercise across Donaldson's 14 chapters. Defer unless priority-1+2 expands.
Donaldson's ~80 exercises vs. Codex's templated 7-block. Dedicated Donaldson-exercise-pack pass (06.E1/06.E2) is P3 follow-up after the theorem-layer batch closes.
Forster-distinctive sheaf machinery beyond Čech (acyclicity, Leray, Mittag-Leffler / Schwartz finiteness). Forster gets its own plan; Čech overlap owned by item 1.
Miranda-distinctive algebraic-curves material (linear systems in full generality, projective embeddings, hyperelliptic theory in detail). Miranda gets its own plan; items 18-19 cover the overlap.
Farkas-Kra-distinctive function-theoretic / theta-deep machinery (Riemann theta with characteristics, Schottky-Klein prime form, Bergman kernel, automorphic forms on Fuchsian groups). Farkas-Kra gets its own plan.
Several-complex-variables material in 06.07-*. Donaldson is strictly one-variable; existing Hörmander / Krantz coverage is fine.
Higher-genus moduli theory beyond Donaldson's Ch. 13 pointer (Teichmüller depth, Mumford GIT, Deligne-Mumford, Mumford-Tate). Hubbard / Mumford / Farb-Margalit territory; item 21 is pointer-stub.
Gauge-theory downstream (Hitchin moduli, Narasimhan-Seshadri). Donaldson's other books; out of scope here.
Worked-example densification across the existing 21 units — curriculum-wide cleanup issue. Only priority-1/2 deepenings carry mandatory rewrites; item 18 addresses systemically.
Chapter-wide $M / X /Σ$ notation cleanup — maintenance pass.

§6 Acceptance criteria for FT equivalence (Donaldson)

Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 and §9, the book is at equivalence-coverage when:

≥95% of Donaldson's named theorems map to Codex units at Donaldson-equivalent proof depth (currently ~35%; after priority-1 this rises to ~70%; after priority-1+2 to ~93%; after priority-3 deepenings to ~96%).
≥80% of Donaldson's exercises have a Codex equivalent (currently ~5% — the templated 7-problem block; closing this requires the dedicated Donaldson-exercise-pack pass per §5).
≥90% of Donaldson's worked examples are reproduced in some Codex unit (currently ~40%; the priority-1+2 batch's required worked-example rewrites bring this to ~65%; item 18 — the running-examples unit — brings this to ~90%).
The notation alignment is recorded inline in the rewritten Master sections (no separate notation/donaldson.md needed).
For every chapter dependency in Donaldson (Part II Ch. 5 → Ch. 6 → Ch. 7; Ch. 8 → Ch. 9 → Ch. 10; Ch. 11 → Ch. 12 → Ch. 13), there is a corresponding prerequisites arrow chain in Codex's DAG. The Hodge-decomp → Serre-duality → Riemann-Roch chain in particular must be unbroken after the priority-1 batch.
Pass-W weaving connects the new units (06.04.02-06, 06.06.06-09, 06.08.01-02, 06.05.04, 06.03.04) to the existing 06.04-riemann-roch-rs/, 06.05-divisors-bundles/, 06.06-jacobians/, and 06.03-riemann-surfaces/ units via lateral connections.

The 7 priority-1 items (items 1-7) close the load-bearing-PDE-and- cohomology-infrastructure gap and the signature-theorem-proof gap. The 10 priority-2 items (items 8-17) close the Donaldson-distinctive depth gaps (bilinear relations, Schottky, Gauss-Manin, VHS) and the deepenings on existing line-bundle / divisor / period-matrix / holomorphic-1-form units. The 6 priority-3+4 items (items 18-23) are depth-completion, examples, and survey pointers; they bring proof-depth coverage from ~93% to ~96%+ but are not strictly required for sign-off.

Composite Donaldson + Forster batch (restated). Donaldson Ch. 8-9 and Forster Ch. III-V overlap on Čech cohomology + Riemann-Roch. Producing both books' priority-1 batches together yields a ~10-unit composite closing the core cohomology gaps simultaneously.

Honest scope. Mid-sized equivalence gap: chapter is already 21 units, topic coverage ~70%. Work dominated by depth rewrites and infrastructure (Čech, Serre duality, Hodge decomposition, $\overset{ˉ}{\partial}$ PDE), not topic-level new content. New units concentrated in cohomology infrastructure (06.04.02-06 — absent) and the moduli / variation block (06.08.* — absent). Jacobian-side units (06.06.06-09) extend an existing section. Line-bundle / divisor / Riemann-Roch units are deepenings of templated v0.5-Strand-B prose.

Largest single Donaldson-distinctive gap: Gauss-Manin / VHS block (items 10-11). Codex has zero coverage; foundational for any future algebraic-geometry / Hodge-theory expansion. Producing items 10-11 is the single largest topic-level expansion Donaldson buys for Codex.

Donaldson — *Riemann Surfaces* (Fast Track 1.07) — Audit + Gap Plan