Forster — *Lectures on Riemann Surfaces* (Fast Track 1.07-paired) — Audit + Gap Plan

Book: Otto Forster, Lectures on Riemann Surfaces (Springer GTM 81, 1981; corrected hardcover 1991; trans. Gilligan from German Riemannsche Flächen, Springer 1977). ISBN 0-387-90617-7.

Fast Track entry: 1.07 (Riemann-surfaces, one-complex-variable; paired with Donaldson, Miranda, Farkas-Kra). Forster is the classical analytic / sheaf-theoretic anchor — distinct from Donaldson's PDE / bundle-theoretic and Miranda's algebraic-curves perspectives.

Purpose: lightweight P1-lite + P2 + P3-lite pass per orchestration protocol — punch-list of new units + deepenings to reach docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4 threshold. Peer audit to donaldson-riemann-surfaces.md; overlap with Donaldson plan flagged so the campaigns share infrastructure rather than double-count.

Codex's 06-riemann-surfaces ships 22 units (just-shipped 06.04.04 Serre duality already cites Forster §17). Topic-level coverage of Forster's core is high; real gaps are on Forster's distinctively analytic spine — Mittag-Leffler on Riemann surfaces, additive and multiplicative Cousin problems, Stein-manifold theory (Theorem A/B in dimension 1), Behnke-Stein for non-compact surfaces, Runge approximation on Riemann surfaces, branched-covering category, $H^1$-finiteness via Schwartz.

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§1 What Forster's book is for

The canonical analytic / sheaf-theoretic introduction in the Cartan-Serre-Behnke-Grauert tradition. Donaldson develops via PDE on a Riemannian manifold (Hodge star, harmonic forms, Dirichlet energy); Miranda via algebraic curves (divisors, linear systems, embeddings); Forster via sheaves of holomorphic functions, Čech cohomology, and the Behnke-Stein existence theorems. ~250 pages, three chapters:

• Chapter 1 — Covering Spaces (§§1-10): Riemann surfaces from gluing patches, holomorphic / meromorphic functions, inverse function theorem on a Riemann surface, general existence theorem for analytic continuation, branched and unbranched coverings, deck transformation groups, universal cover, Galois coverings, Riemann's existence theorem for $P(z,w)=0$.
• Chapter 2 — Compact Riemann Surfaces (§§11-22): the cohomology machinery. Differential forms, Čech cohomology, the finiteness theorem $\dim H^1(X,\mathcal{O})<\infty$ via Schwartz's compact-operator theorem, divisors, Riemann-Roch $\dim L(D)-\dim I(D)=1-g+\mathrm{deg}D$, Serre duality $I(D)\cong L(K-D{)}^{\ast }$, Abel, Jacobi inversion, theta functions.
• Chapter 3 — Non-Compact Riemann Surfaces (§§23-31): the Forster-distinctive Behnke-Stein-Cartan-Serre block. Cousin I/II, Runge approximation on RS, Theorems A and B for Stein RS, the Behnke-Stein theorem (non-compact ⇒ Stein) — peculiar to Forster among the FT 1.07 quartet — and Mittag-Leffler on RS as apex.

Distinctive Forster choices:

1. Sheaf-theoretic from §11; no scheme theory. Coherent analytic sheaves + Čech as working language; never enters Hartshorne-style AG. Distinct from a scheme audience.
2. Cousin I + II (§§26-27). Codex has zero coverage.
3. Mittag-Leffler on Riemann surfaces (§26/§28) — analytic apex of Chapter 3; not headline in Donaldson, Miranda, or Farkas-Kra.
4. Cohomology of Stein manifolds (1-d) — Theorems A and B (§§28-30). Donaldson skips Stein theory entirely.
5. Behnke-Stein theorem on non-compact RS (§29). Non-compact ⇒ Stein. Largest Forster-distinctive gap in Codex.
6. Rigorous gluing-patches foundation (§§1-2). More explicit than Donaldson — chart-atlas, IFT on RS, maximal atlas. Codex unit doesn't unpack at this level.
7. Branched coverings as a category (§§4-6). Branched extensions of $X^{\prime }\to Y\setminus B$ via local-monodromy criterion; identification of ${\mathbb{P}}^1$ branched covers with finite extensions of $\mathbb{C}(z)$. Not in Codex.
8. Riemann's existence theorem (§§7-8). Every compact RS arises from $P(z,w)=0$, via Mittag-Leffler. The concrete complex-analytic ↔ algebraic-curve bridge.
9. Schwartz finiteness for $H^1(X,\mathcal{O})$ (§§14-15). Banach compact-perturbation argument; Donaldson uses Hodge instead.
10. Mittag-Leffler / Cousin I / II as one arc (§§26-28). The climax-narrative of Chapter 3 as Theorem B applied to the principal-part sheaf — Forster's pedagogical contribution.
11. 254 pages, ~50 exercises. Less ground than Donaldson but covers non-overlapping analytic spine Donaldson skips entirely.

Forster does not cover: PDE uniformisation, Hodge decomposition, Gauss-Manin / VHS, Schottky, projective-embedding criteria beyond Riemann's existence — all Donaldson territory.

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§2 Coverage table (Codex vs Forster)

Cross-referenced against the 22-unit Codex Riemann-surfaces chapter, listed below. ✓ = covered at Forster-equivalent depth, △ = topic present but Codex unit shallower (typically templated v0.5 Strand-B prose), ✗ = not covered.

Existing Codex units (find content/06-riemann-surfaces -name "*.md" | sort): 06.01.01-holomorphic-function, 06.01.02-cauchy-integral-formula, 06.01.03-residue-theorem, 06.01.04-analytic-continuation, 06.01.05-meromorphic-function, 06.01.06-riemann-mapping-theorem, 06.02.01-branch-point-ramification, 06.03.01-riemann-surface, 06.03.02-genus-riemann-surface, 06.03.03-uniformization-theorem, 06.04.01-riemann-roch-compact-riemann-surfaces, 06.04.04-serre-duality-curves (just-shipped, Forster §17 cited), 06.05.01-divisor-riemann-surface, 06.05.02-holomorphic-line-bundle, 06.05.03-riemann-hurwitz-formula, 06.06.01-holomorphic-one-form, 06.06.02-period-matrix, 06.06.03-jacobian-variety, 06.06.04-abel-jacobi-map, 06.06.05-theta-function, 06.07.01-holomorphic-several-variables, 06.07.02-hartogs-phenomenon.

Chapter 1 — Covering Spaces (§§1-10)

Forster topic	Codex unit	Status	Note
§1 Definition of RS (chart-atlas, gluing patches)	06.03.01-riemann-surface	△	Definition shipped; Forster's explicit gluing-patches (maximal atlas, equivalence) implicit. Mild depth gap.
§1 Examples — ${\mathbb{P}}^1$, tori, $P(z,w)=0$, hyperelliptic	partial	△	Scattered in Master sections; no running-examples unit (overlap with Donaldson item 18).
§2 Holomorphic / meromorphic functions on RS	06.01.01, 06.01.05	✓	Shipped at depth.
§3 Inverse function theorem on RS; identity	partial in 06.01.04	△	Forster's local-form-of-holomorphic-map theorem implicit.
§4 Branch points / ramification	06.02.01-branch-point-ramification	✓	Shipped. Forster citation would strengthen tier_anchors.
§§4-6 Branched coverings as a category	—	✗	Gap. Functorial / categorical construction of branched extensions absent. Branch-point unit covers local picture only.
§6 Galois / deck-transformation theory	partial in 06.03.03 Master	△	Universal cover / monodromy mentioned; Galois / deck-group treatment not unpacked.
§7 General analytic-continuation existence theorem	partial in 06.01.04	△	Forster's monodromy-along-paths formulation not unpacked.
§8 Riemann's existence theorem ($P(z,w)=0$ ↔ compact RS; finite extensions of $\mathbb{C}(z)$)	—	✗	Gap (priority-2/3). Forster-signature; every compact RS is the normalisation of a plane algebraic curve.
§§9-10 Algebraic functions, monodromy of $P(z,w)=0$	—	✗	Pointer to Riemann's existence.

Chapter 2 — Compact Riemann Surfaces (§§11-22)

Forster's sheaf-theoretic spine begins. Heavy overlap with Donaldson Part II — most items co-owned in the composite batch.

Forster topic	Codex unit	Status	Note
§11 Differential forms on RS	partial in 06.06.01	△	Composite — same gap as Donaldson (Hodge-star / $(p,q)$-form vocabulary).
§12 1-form integration; periods	06.06.02-period-matrix	△	Composite — Donaldson item 16.
§13 Sheaves on RS; Čech cohomology	—	✗	Gap (high priority). Co-owned with Donaldson item 1. One unit serves both.
§14 Schwartz finiteness $\dim H^1(X,\mathcal{O})<\infty$	—	✗	Gap (high priority — Forster-distinctive). Donaldson proves via Hodge (Donaldson item 2); Forster's compact-operator proof is not co-owned.
§16 Divisors + Riemann-Roch (Forster's proof)	06.04.01	△	Composite — Donaldson item 4.
§17 Serre duality (residue pairing)	06.04.04-serre-duality-curves (just shipped)	✓	Covered. Forster §17 cited; serves both books.
§18 Riemann-Hurwitz	06.05.03-riemann-hurwitz-formula	✓	Shipped.
§§19-20 $\dim {\Omega }^1(X)=g$	06.06.01	△	Composite — Donaldson item 17.
§21 Abel's theorem	06.06.04-abel-jacobi-map	✓	Shipped.
§§21-22 Jacobi inversion (theta proof)	—	✗	Composite — Donaldson item 6.
§22 Theta functions, vanishing theorem	06.06.05-theta-function	△	Composite — Donaldson item 7; vanishing not proved.

Chapter 3 — Non-Compact Riemann Surfaces (§§23-31)

Forster's distinctive territory. Donaldson skips this entire block. Zero overlap with Donaldson plan; all gaps owned by Forster plan.

Forster topic	Codex unit	Status	Note
§23 Sheaf cohomology on non-compact spaces; fine-sheaf acyclicity	—	✗	Gap. Foundation for §§24-30.
§24 Dolbeault iso $H^1(X,\mathcal{O})=H_{\bar{\partial }}^{0,1}(X)$	—	✗	Gap. Sheaf $H^1$ ↔ $\bar{\partial }$.
§§25-26 Cousin I (additive) on non-compact RS	—	✗	Gap (high priority — Forster-distinctive). Local principal-parts data ⇒ global meromorphic function.
§26 Cousin I cohomologically: $H^1(X,\mathcal{O})=0\Rightarrow$ solvable	—	✗	Gap. Same unit.
§27 Cousin II (multiplicative)	—	✗	Gap (high priority — Forster-distinctive). Prescribed divisors; obstruction in $H^2(X,\mathbb{Z})$.
§27 Weierstrass product theorem on non-compact RS	—	✗	Gap. Cousin II corollary.
§28 Mittag-Leffler on non-compact RS	—	✗	Gap (high priority — Forster-distinctive headline). Apex of Theorem B.
§29 Theorem A	—	✗	Gap (high priority). Cartan-Serre A in dim 1.
§29 Theorem B ($H^q(X,\mathcal{F})=0$, $q\ge 1$, Stein, coherent)	—	✗	Gap (high priority). Headline vanishing; Cousin / Mittag-Leffler derive.
§29 Stein Riemann surface definition	—	✗	Gap. Definition unit.
§29 Behnke-Stein theorem (non-compact ⇒ Stein)	—	✗	Gap (highest priority — Forster-distinctive headline). Foundation for the Cartan-Serre chain on RS.
§30 Runge approximation on RS	—	✗	Gap (high priority). Generalises Runge on $\mathbb{C}$.
§31 Line bundles trivial on non-compact RS	partial in 06.05.02	△	Cousin-II corollary; Master-section addition worth adding.

Aggregate coverage estimate (vs Forster)

• Theorem layer: ~55% topic, ~30% Forster-proof-depth. Gap in Chapter 3 (entire Stein / Cousin / Behnke-Stein / Mittag-Leffler / Runge block, zero coverage) + Schwartz finiteness + Riemann's existence + branched-covering category. After priority-1: topic ~90%, proof-depth ~70%. After priority-1+2: ~92%.
• Exercise layer: ~50 Forster exercises vs templated 7-block. Composite with Donaldson exercise pack, deferred.
• Worked-example layer: ~45%. Running examples overlap Donaldson; composite item (Donaldson item 18).
• Notation layer: ~75% aligned. Forster $X$, $\mathcal{O}$, $\mathcal{M}$, $L(D)=H^0(\mathcal{O}(D))$, $I(D) = H^1(\mathcal{O}(D))$.$L(D)/I(D)$ Forster-specific.
• Sequencing layer: ~60%. Codex DAG matches Chapter-2 flow; Chapter-3 chain (non-compact sheaf ⇒ Theorem B ⇒ Cousin ⇒ Mittag-Leffler) absent.
• Intuition layer: ~50%. Sheaf intuition partial; Stein intuition (holomorphic convexity, why obstructions vanish on non-compact) absent.
• Application layer: ~50%. Mittag-Leffler / Weierstrass on $\mathbb{C}$ implicit in 06.01.05; RS generalisations absent.

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

Codex covers Forster Chapters 1-2 reasonably at topic level; the Donaldson plan handles depth deepenings on the composite block. Forster's distinctive contribution to Codex is Chapter 3 — the Stein / Cousin / Behnke-Stein / Mittag-Leffler / Runge block. New units open 06.04-cohomology/ (already opened by 06.04.04) and a new 06.09-stein-and-cousin/.

Overlap convention. Items tagged [Forster-only] (sole-owned by this plan), [Composite] (also in Donaldson plan; one unit serves both — production accounted to Donaldson). Composite items listed with cross-reference, not double-counted.

Priority 1 — Forster-distinctive analytic-spine (Chapter 3 block)

Forster's signature pedagogical contribution. Donaldson plan covers none of these. Without them Codex cannot claim Forster-equivalence.

1. 06.09.01 Stein Riemann surfaces. [Forster-only] Forster §29; Cartan-Thullen 1932 / Stein 1951 originators; Grauert 1958 for higher-dim. ~1800 words. Master: holomorphic convexity; equivalence on RS (non-compact ⇒ Stein); $\mathcal{O}(X)$ as Fréchet algebra; setup for Theorem A/B. Foundational for items 2-5.

2. 06.09.02 Theorem A and B for Stein RS. [Forster-only] Forster §29; Cartan séminaire 1951-52, Cartan-Serre 1953 (CRAS 237). ~2200 words. Master: A — coherent sheaves generated by global sections; B — $H^q(X,\mathcal{F})=0$ for $q\ge 1$. Forster's 1-d proof: exhaustion by relatively-compact Runge opens

   • Schwartz finiteness + limit passage. Worked: $X=\mathbb{C}$, $\mathcal{F}=\mathcal{O}$, $H^1=0$ recovers classical Mittag-Leffler. Headline of Forster-distinctive block.

3. 06.09.03 Behnke-Stein on non-compact RS. [Forster-only] Forster §29; Behnke-Stein 1949 (Math. Ann. 120) originator. ~1800 words. Master: every non-compact connected RS is Stein; proof via Behnke-Stein exhaustion (concentric Runge domains); consequence chain (line bundles trivial; Cousin I/II solvable; Mittag-Leffler holds). Largest Forster gap; foundational for 4-7.

4. 06.09.04 Cousin I (additive). [Forster-only] Forster §§25-26; Cousin 1895 (Acta Math. 19). ~1500 words. Master: data $(U_i,f_i)$ with $f_i-f_j\in \mathcal{O}(U_i\cap U_j)$ → global $f\in \mathcal{M}(X)$ with $f{|}_{U_i}-f_i\in \mathcal{O}$. Obstruction in $H^1(X,\mathcal{O})$; vanishes non-compact.

5. 06.09.05 Cousin II (multiplicative). [Forster-only] Forster §27; Cousin 1895. ~1500 words. Master: prescribed local divisors. Exponential sequence $0 \to \mathbb{Z} \to \mathcal{O} \to \mathcal{O}^* \to 0$\to obstructionin$H^2(X, \mathbb{Z})$; non-compact ⇒ trivial → Cousin II solvable; line bundles trivial.

6. 06.09.06 Mittag-Leffler on RS. [Forster-only] Forster §28; Mittag-Leffler 1884 plane case; Behnke-Stein 1949 RS case. ~1700 words. Master: plane case as warm-up; RS generalisation (prescribed principal parts realised by global meromorphic function) via Cousin I. Worked: $\mathbb{C}$, ${\mathbb{C}}^{\ast }$. Apex unit; Forster's headline.

7. 06.09.07 Runge approximation on RS. [Forster-only] Forster §30; Runge 1885; Behnke-Stein 1949 RS extension. ~1500 words. Master: classical Runge lifted to non-compact RS; setup for items 2-3, 6 exhaustion arguments.

Priority 2 — Forster-distinctive proofs + covering-space block

Forster-distinctive proofs of theorems already (or soon-to-be) covered, plus the branched-covering / Riemann-existence block from Chapter 1.

8. 06.04.05 Schwartz finiteness for $H^1(X,\mathcal{O})$ on compact RS. [Forster-only] Forster §§14-15; Schwartz 1950. ~1700 words. Master: Čech cocycles in Banach; obstruction as compact perturbation of injection ⇒ $\dim \mathrm{coker}<\infty$. Forster-distinctive proof; Donaldson uses Hodge (item 2).

9. 06.02.02 Branched coverings as a category. [Forster-only] Forster §§4-6; Riemann 1851 / Klein 1882. ~1800 words. Master: unbranched cover $X^{\prime }\to Y\setminus B$; branched-extension theorem (unique extension if local monodromy finite); deck / Galois covers. Worked: $\sqrt{z}$ extends to double cover of ${\mathbb{P}}^1$ branched at $\{0,\infty \}$.

10. 06.02.03 Riemann's existence theorem for algebraic curves. [Forster-only] Forster §§7-8; Riemann 1857 (Theorie der Abel'schen Functionen). ~2000 words. Master: every compact RS arises as desingularisation of $P(z,w)=0$; $\mathcal{M}(X)$ finite over $\mathbb{C}(z)$; categorical equivalence compact-RS ↔ smooth-projective-curves. Depends on item 6. Conceptual climax of Chapter 1.

11. 06.04.06 Dolbeault iso $H^1(X, \mathcal{O}) = H^{0,1}_{\bar\partial}(X)$. [Forster-only] Forster §24; Dolbeault 1953 (CRAS 236). ~1500 words. Master: ${\mathcal{E}}^{0,1}$ fine ⇒ resolution gives Dolbeault iso. Bridges sheaf cohomology (Donaldson 1 / Forster §13) to Hodge (Donaldson 2).

12. Deepen 06.03.01-riemann-surface (gluing patches). [Forster-only] Add Forster §§1-3 explicit chart-atlas: inverse function theorem on RS, maximal atlas, equivalent atlases, local-form-of-holomorphic-map theorem.

13. Deepen 06.05.02-holomorphic-line-bundle (triviality on non-compact RS). [Forster-only] Forster §31 — every holomorphic line bundle on non-compact RS trivial (corollary of Cousin II / item 5).

Priority 3 — composite-batch overlap with Donaldson (cross-references)

14. 06.04.02 Čech cohomology of analytic sheaves on RS. [Composite — Donaldson item 1.] Forster §13. Production owned by Donaldson plan. Forster framing adds analytic-sheaf vocabulary ($\mathcal{O},\mathcal{M},{\mathcal{O}}^{\ast },\mathcal{O}(D)$) and the obstruction-class interpretation Chapter 3 needs.

15. Deepen 06.04.01-riemann-roch (Forster §16 proof). [Composite — Donaldson item 4.] Same divisor-bumping induction

    • Serre duality proof; cite both books.

16. Deepen 06.06.05-theta-function and produce Jacobi inversion 06.06.06. [Composite — Donaldson items 6-7.] Forster §22.

17. Deepen 06.06.01-holomorphic-one-form (Hodge dim-counting). [Composite — Donaldson item 17.] Forster §§19-20.

18. Deepen 06.05.03-riemann-hurwitz-formula (Forster §18 proof). [Composite] Forster's combinatorial / branched-cover counting proof as alternative-proof commentary. Low priority.

19. Deepen 06.06.02-period-matrix for Forster §12 framing. [Composite — Donaldson item 16].

Priority 4 — Survey / pointer items, optional

20. 06.09.08 Survey: Cartan-Serre Stein theory in higher dim. [Forster-only — pointer] Master-only; ~700 words. Pointer to Hörmander Ch. IV-VI / Grauert-Remmert Theory of Stein Spaces.

21. Notation crosswalk (notation/forster.md). [Forster-only] Forster-specific $L(D)/I(D)$ + sheaf names. ~400 words. Optional.

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§4 Implementation sketch (P3 → P4); overlap with Donaldson plan

Minimum Forster-equivalence batch = priority 1 (items 1-7): 7 new units, all Forster-only (zero overlap with Donaldson plan). Production estimate (mirroring Donaldson / Cannas batches):

• ~3 h per typical new unit; ~4 h each for items 2, 3 (large, load-bearing).
• Priority 1: 2 × 4 + 5 × 3 = ~23 h.
• Priority 1+2: 23 h + 4 typical × 3 h + 2 deepenings × 1 h = ~37 h.
• Priority 3 items (14-19) are deepenings already in Donaldson plan — producing Donaldson priority-1 closes them as a side-effect. Central scheduling insight.

At 4-6 parallel production agents, priority-1 fits in a 2-day window.

Composite Donaldson + Forster scheduling. Recommended order:

• Phase Composite-A (Donaldson priority-1, ~14 h): Čech cohomology, Hodge decomp, Serre duality (already shipped), Hilbert PDE, main theorem; deepens Riemann-Roch and uniformization. Closes Forster priority-3 items 14-19.
• Phase Forster-1 (~23 h): Chapter 3 block, items 1-7. All Forster-only; zero Donaldson overlap. Run after Composite-A so Theorem B (item 2) cites the Čech unit without forward-declaration.
• Phase Forster-2 (~12 h): items 8-11. Item 8 cross-references Donaldson item 2; item 10 depends on item 6.
• Phase Forster-3 (~3 h): deepenings 12, 13.

Batches within Forster Phase 1.

• F-A (items 1, 2, 3, ~10 h): Stein foundation; opens 06.09-stein-and-cousin/. Load-bearing.
• F-B (items 4, 5, 6, 7, ~12 h): Cousin / Mittag-Leffler / Runge, depends on F-A.

Originator-prose targets (each Forster-distinctive unit cites originator + Forster):

• (1) Stein RS: Cartan-Thullen 1932 (Math. Ann. 106); Stein 1951; Grauert 1958.
• (2) Theorem A/B: Cartan séminaire 1951-52; Cartan-Serre 1953 (CRAS 237); Serre 1955 Faisceaux algébriques cohérents (Ann. Math. 61) for the algebraic parallel.
• (3) Behnke-Stein 1949 (Math. Ann. 120, 430-461). Originator voice mandatory.
• (4, 5) Cousin I/II: Cousin 1895 (Acta Math. 19); H. Cartan 1934.
• (6) Mittag-Leffler 1884 (Acta Math. 4) plane case; Behnke-Stein 1949 for RS.
• (7) Runge 1885 (Acta Math. 6); Behnke-Stein 1949 RS extension.
• (8) Schwartz 1950 (Bull. Sci. Math. 74).
• (9) Branched covers: Riemann 1851 Grundlagen (Inauguraldissertation); Klein 1882.
• (10) Riemann's existence: Riemann 1857 Theorie der Abel'schen Functionen (J. reine angew. Math. 54). Originator voice mandatory.
• (11) Dolbeault 1953 (CRAS 236).

Notation crosswalk. Small notation/forster.md for $L(D)/I(D)$, choice of $X$ (vs Donaldson's $M$), and sheaf names. Defer unless chapter-wide cleanup (item 21).

DAG edges to add (priority-1+2):

• 06.09.01 ← {06.03.01, 06.04.02-cech-cohomology (Donaldson)}
• 06.09.02 ← {06.09.01, 06.04.02, 06.09.07}
• 06.09.03 ← {06.09.01, 06.09.07}
• 06.09.04 ← {06.09.02, 06.09.03}
• 06.09.05 ← {06.09.02, 06.09.03, 06.05.02}
• 06.09.06 ← {06.09.04, 06.01.05}
• 06.09.07 ← {06.01.06, 06.09.01}
• 06.04.05 (item 8) ← {06.04.02}; xref 06.04.03-hodge
• 06.02.02 (item 9) ← {06.02.01, 06.03.01}
• 06.02.03 (item 10) ← {06.02.02, 06.09.06}
• 06.04.06 (item 11) ← {06.04.02, 06.06.01}

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§5 What this plan does NOT cover

• Line-number-level inventory of every theorem / exercise across Forster's 31 sections. Defer.
• Forster's ~50 exercises vs. Codex templated 7-block. Composite with Donaldson exercise pack — P3 follow-up.
• Donaldson-distinctive material (PDE uniformisation; Hodge decomp; Gauss-Manin / VHS; Schottky; projective-embedding criteria beyond Riemann's existence). Donaldson plan owns.
• Miranda-distinctive algebraic-curves material (full linear- systems, hyperelliptic theory). Miranda plan owns.
• Farkas-Kra-distinctive function-theoretic / theta-deep machinery (theta with characteristics, Schottky-Klein prime form, Bergman kernel, automorphic forms). Farkas-Kra plan owns.
• Several-complex-variables Stein theory (Hörmander, Grauert-Remmert). Forster is dimension 1; higher-dim is item 20 pointer.
• Composite-block accounting. Items 14-19 are cross-references to Donaldson; production owned by Donaldson plan. Read this Forster plan as adding ~13 units to Codex (priority-1: 7 + priority-2: 4 new + 2 deepenings), not 21.
• Worked-example densification beyond priority-1+2 deepenings.
• Chapter-wide $M/X/\Sigma$ notation cleanup — joint maintenance pass; out of scope.
• Higher-genus moduli theory (Donaldson Ch. 13 / Mumford GIT territory).
• Gauge theory / Hitchin moduli downstream.

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§6 Acceptance criteria for FT equivalence (Forster)

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

• ≥95% of Forster's named theorems map to Codex units at Forster-equivalent proof depth (currently ~30%; after priority-1, ~75%; after priority-1+2, ~92%; with composite Donaldson priority-1 in parallel, ~96%).
• ≥80% of Forster's exercises have a Codex equivalent (currently ~5%; composite Donaldson+Forster exercise pack closes this).
• ≥90% of Forster's worked examples reproduced in some Codex unit (currently ~45%; priority-1+2 worked-example sections + composite running-examples unit (Donaldson item 18) bring this to ~90%).
• Notation alignment recorded inline in priority-1 Master sections plus optional notation/forster.md (item 21).
• DAG prerequisites arrows for every Forster chapter dependency (§13 → §16; §13 → §17; §29 → §28 → §27 → §25; §30 → §29). Behnke-Stein → Theorem B → Cousin → Mittag-Leffler chain unbroken after priority-1.
• Pass-W weaving connects new units (06.09.01-07, 06.04.05-06, 06.02.02-03) to existing chapter — especially to the just-shipped 06.04.04 Serre-duality (which cites Forster §17), 06.05.02 line-bundle (becomes trivial on non-compact RS via item 5), and 06.01.05 meromorphic-function (lifts to global existence via item 6).

Priority-1 (items 1-7) closes the Forster-distinctive analytic spine (Stein / Cousin / Behnke-Stein / Mittag-Leffler / Runge — all of Chapter 3). Priority-2 new units (8-11) + 2 deepenings (12-13) close the Forster-distinctive proof-style gap (Schwartz finiteness, branched-covering category, Riemann's existence, Dolbeault) and gluing-patches / non-compact-line-bundle deepenings. Priority-3 (14-19) is composite-batch overlap with Donaldson — not double- counted. Priority-4 (20-21) optional pointer / notation crosswalk.

Scheduling. Run Donaldson Phase Composite-A first (~14 h — Čech / Hodge / Serre-duality (shipped) / Riemann-Roch infrastructure). Then Forster Phase 1 (~23 h — Chapter 3 block, opening 06.09-stein-and-cousin/). Composite priority-1 cost: ~37 h, ~12 new units across both books + ~5 deepenings. Priority-2 blocks (Gauss-Manin / VHS; Schwartz / Riemann's existence / Dolbeault; ~20 h combined) round out FT-1.07 near-completely on both anchors.

Honest scope. Mid-sized gap on a 22-unit chapter. Work dominated by Forster's distinctive Chapter 3 analytic spine (zero current Codex coverage), plus composite infrastructure shared with Donaldson (Čech, Riemann-Roch proof, Serre duality — partly shipped). New Forster-only units concentrate in 06.09-stein-and-cousin/ (items 1-7) plus covering-space supplements (8-11). Composite batching saves ~7-10 h of duplicate Čech / Riemann-Roch work.

Largest Forster-distinctive gap: Behnke-Stein (item 3) and the chain Theorem A/B → Cousin I/II → Mittag-Leffler on RS. Items 1-7 are the single largest topic-level expansion Forster buys Codex; no other FT-1.07 anchor (Donaldson, Miranda, Farkas-Kra) supplies them.