Voisin — *Hodge Theory and Complex Algebraic Geometry I* (Fast Track 3.27) — Audit + Gap Plan

Book: Claire Voisin, Hodge Theory and Complex Algebraic Geometry, I (translated by Leila Schneps). Cambridge Studies in Advanced Mathematics 76, Cambridge University Press 2002. xi + 322 pp. (Translation of Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, SMF 2002.) ISBN 0-521-80260-1. The canonical modern English-language reference for Hodge theory on smooth projective complex / compact Kähler manifolds: the Kähler-package proofs — harmonic theory, the Kähler identities, the $\partial \bar{\partial }$-lemma, the Hodge decomposition, the Lefschetz decomposition, hard Lefschetz, Hodge-Riemann bilinear relations, the Hodge index theorem — written by an originator of the modern subject and structured as the strict prerequisite for Voisin Vol II (Hodge classes, the Hodge conjecture, mixed Hodge structures, Noether-Lefschetz).

Fast Track entry: 3.27. Listed in docs/catalogs/FASTTRACK_BOOKLIST.md as "Hodge theory via complex geometry" alongside FT 3.28 (Jost, Riemannian Geometry and Geometric Analysis) which serves as the differential-analysis companion. Sibling to FT 3.18 / 3.19 (Kobayashi-Nomizu Vol I + Vol II, audits shipped Cycles 4 + 5 respectively — see plans/fasttrack/kobayashi-nomizu-foundations-vol1.md and plans/fasttrack/kobayashi-nomizu-foundations-vol2.md). The KN-II audit explicitly flagged Voisin Vol I as a forward dependency for the Kähler-Hodge package; this plan picks up that thread.

Purpose of this plan: lightweight audit-and-gap pass (P1-lite + P2 + P3-lite of the orchestration protocol). Output is a concrete punch-list of new units to write so that Hodge Theory and Complex Algebraic Geometry I (VHCAG-I hereafter) is covered to the equivalence threshold (≥95% effective coverage of theorems, key examples, exercise pack, notation, sequencing, intuition, applications — see docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §3.4).

Audit mode: REDUCED. No local PDF is available (reference/textbooks-extra/ and reference/fasttrack-texts/03-modern-geometry/ checked; VHCAG-I is under active CUP / SMF copyright and was not retrieved within the 3-hour time budget). This pass works from the canonical VHCAG-I table of contents (Parts I–III, Chs. 1–11 + appendices, well-attested across the modern Hodge-theoretic literature), the peer-source crosswalks below (Griffiths-Harris, Wells, Huybrechts, Demailly), the Fast Track entry's editorial framing, the in-Codex evidence that VHCAG-I is already cited as a master anchor in at least five shipped units (04.09.01-hodge-decomposition.md, 04.09.02-kodaira-vanishing.md, 04.05.09-hodge-index-theorem.md, 06.04.03-hodge-decomposition-curves.md, 06.04.05-dbar-hilbert-pde.md), and the KN-II audit (Cycle 5) findings for the differential-geometric prerequisite layer. A full P1 line-number inventory is deferred to the production pass when a local copy is on disk.

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§1 What VHCAG-I is for

VHCAG-I is the definitive modern reference for Hodge theory on smooth projective complex manifolds. Where Griffiths-Harris Principles of Algebraic Geometry (1978) Ch. 0 gives the foundational sketch and Wells Differential Analysis on Complex Manifolds (GTM 65, 1980) gives the hard-PDE treatment, Voisin assembles a single self-contained, modern presentation that is simultaneously (i) rigorous about the PDE input (harmonic theory, ellipticity, regularity), (ii) systematic about the Kähler-package theorems and their interrelations, and (iii) explicitly structured as a prerequisite for Voisin Vol II (mixed Hodge structures, Hodge classes, the Hodge conjecture, normal functions, Noether-Lefschetz). The book is written by an originator of the modern subject — Voisin's own work on the Hodge conjecture, the integral Hodge conjecture counterexamples, decomposition of the diagonal, and rational equivalence of zero-cycles forms much of the contemporary Hodge-theoretic landscape that Vol II surveys — and the choices of emphasis throughout Vol I reflect which inputs Vol II will need.

Distinctive content, organised by the three parts of the volume:

1. Part I (Chs. 1–4) — Preliminaries: complex manifolds and holomorphic bundles. Complex manifolds, holomorphic vector bundles, sheaves and sheaf cohomology, connections and curvature, Chern classes via curvature. Voisin's Part I is denser and more efficient than the parallel material in Griffiths-Harris Ch. 0 or Wells Chs. I–III, with explicit signposting of which preliminaries are load-bearing for the Hodge-theoretic core. The Dolbeault complex $({\Omega }^{p,\bullet },\bar{\partial })$ and the Dolbeault isomorphism $H^q(X;{\Omega }^p)\cong H_{\bar{\partial }}^{p,q}(X)$ are developed here as the bridge between sheaf cohomology and the later harmonic-form representatives [ref: D. Huybrechts, Complex Geometry: An Introduction (Universitext, Springer 2005) Ch. 2 — the modern accessible parallel; R. O. Wells, Differential Analysis on Complex Manifolds (GTM 65, Springer 3rd ed. 2008) Chs. I–III — the classical hard-analysis parallel].
2. Part II (Chs. 5–7) — Harmonic theory and the Hodge decomposition. The technical heart of the volume. Ch. 5 develops the PDE foundation: the formal adjoint $d^{\ast }$ on a compact Riemannian manifold, the Laplace-Beltrami operator $\Delta =d{d}^{\ast }+d^{\ast }d$ as a self-adjoint elliptic operator, the Hodge theorem (every de Rham class has a unique harmonic representative), the Hodge orthogonal decomposition $\Omega^k(X) = \mathcal{H}^k \oplus d\Omega^{k-1} \oplus d^\Omega^{k+1}$. Voisin gives the PDE input explicitly — Gårding's inequality, elliptic regularity, Fredholm theory of self-adjoint elliptic operators on compact manifolds — rather than black-boxing it as Griffiths-Harris does. Ch. 6 develops the Kähler identities $[\Lambda, \bar\partial] = -i\partial^$,$[\Lambda, \partial] = i\bar\partial^$ and their consequence ${\Delta }_d=2{\Delta }_{\partial }=2{\Delta }_{\bar{\partial }}$ on a Kähler manifold, then the Hodge decomposition $H^k(X; \mathbb{C}) = \bigoplus_{p + q = k} H^{p, q}(X)$ on a compact Kähler manifold, the Hodge symmetry $\overline{H^{p,q}}=H^{q,p}$, and the $\partial \bar{\partial }$-lemma: on a compact Kähler manifold, every $d$-exact $(p,q)$-form is $\partial \bar{\partial }$-exact. Ch. 7 develops the Lefschetz decomposition: with $L=\omega \wedge$ and $\Lambda$ its adjoint, the operators $(L,\Lambda ,H)$ form an ${\mathfrak{sl}}_2$ action on $H^(X; \mathbb{C})$, giving the primitive decomposition $H^k={⨁}_r{L}^r{P}^{k-2r}$ and the hard Lefschetz theorem $L^k:H^{n-k}(X)\stackrel{\sim }{\to }{H}^{n+k}(X)$. The Hodge-Riemann bilinear relations on primitive cohomology give positivity, leading to the Hodge index theorem. This is the canonical statement of the Kähler package; the K3-surface signature analog from Hartshorne Algebraic Geometry V Theorem 1.9 is the surface-Néron-Severi specialisation [ref: P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley 1978) Ch. 0 §6–§7 — the classical sketch parallel; J.-P. Demailly, Complex Analytic and Differential Geometry (OpenContent monograph, current online edition) Chs. VI + VIII — the most rigorous PDE-side parallel].
3. Part III (Chs. 8–11) — Applications and structure theory. Ch. 8 develops the Lefschetz $(1,1)$-theorem identifying $\mathrm{NS}(X) \otimes \mathbb{R}$with$H^{1, 1}(X; \mathbb{R}) \cap H^2(X; \mathbb{Z})$ on a smooth projective $X$; Ch. 9 develops Kodaira vanishing ($H^q(X;K_X\otimes L)=0$ for $L$ ample and $q>0$) via the Akizuki-Nakano-Bochner identity on a Kähler manifold; Ch. 10 develops Kodaira's embedding theorem (compact Kähler $X$ with an integral Kähler class is projective) — the bridge from Kähler manifolds to projective algebraic varieties; Ch. 11 develops the basic Hodge structures on $H^k$ as a setup for Vol II [ref: Huybrechts §3.3 + §5; Demailly Ch. VII; K. Kodaira, "On Kähler varieties of restricted type," Annals of Math. 60 (1954) 28–48 — the originating paper for the embedding theorem]. Voisin Vol II then takes off from these — mixed Hodge structures on non-compact / singular varieties, Hodge classes and the Hodge conjecture, variations of Hodge structure and the period map, Noether-Lefschetz theory.
4. Calabi-Yau / Kähler-Einstein pointer. VHCAG-I touches the Calabi-Yau / Kähler-Einstein literature only by pointer: a compact Kähler manifold with $c_1(K_X)=0$ admits a Ricci-flat Kähler metric (Yau 1977 — solution of the Calabi conjecture, Comm. Pure Appl. Math. 31, 339–411) and is then called Calabi-Yau in the strict differential- geometric sense. Voisin states this as a pointer to the much more substantial Yau / Aubin / Tian deep PDE theory and defers it; the Codex follows the same convention (see §5 below).
5. Editorial signature. VHCAG-I is uncompromisingly PDE-first inside the Kähler package: every theorem in Part II flows from the harmonic-theoretic ingredient established in Ch. 5, and the PDE input is made explicit rather than imported as a black box. This is the opposite stylistic choice from Griffiths-Harris (which keeps the PDE technical machinery in the background and emphasises the algebraic- geometric corollaries) and the same choice as Wells and Demailly. Among the modern English-language Hodge-theoretic references it is the only one that combines (a) full PDE rigour in the harmonic- theoretic core, (b) a systematic Kähler-package presentation, and (c) the explicit Vol-II-prerequisite framing. Huybrechts Complex Geometry (Universitext 2005) is the accessible-but-less-rigorous alternative; Demailly Complex Analytic and Differential Geometry is the rigorous-but-encyclopedic alternative; Griffiths-Harris is the classical-but-dated alternative. Voisin is the modern default [ref: Huybrechts Preface — "an alternative to Griffiths-Harris and Voisin for readers wanting a less compressed entry"; Demailly Preface — "a comprehensive treatment overlapping with Voisin and Wells, with more weight on the analysis"].
6. Peer-source crosswalk. The Voisin-Griffiths-Harris-Wells-Huybrechts- Demailly cluster is the modern canonical entry to Hodge theory; the Codex Fast Track lists Voisin (3.27) as the recommended primary text. Among these, the peer-source agreement on what the Kähler package is (Hodge decomposition + $\partial \bar{\partial }$-lemma + Lefschetz decomposition + hard Lefschetz + Hodge-Riemann + Kodaira vanishing + Kodaira embedding) is essentially complete; the disagreements are stylistic, pedagogical, and notational.
7. VHCAG-I is not a first textbook on complex manifolds (Huybrechts serves that role at a more accessible pace), not a first textbook on harmonic analysis on manifolds (Warner, Wells, or Jost serve that role), and not a survey of contemporary Hodge theory (Voisin Vol II is that book). The canonical "before VHCAG-I" sequence is Griffiths-Harris Ch. 0 → Wells → VHCAG-I, or Huybrechts → VHCAG-I directly. The canonical "after VHCAG-I" sequence is VHCAG-I → Voisin Vol II → mixed Hodge modules (Saito) or → Donaldson-Kronheimer (four-manifold gauge theory) or → Joyce (special holonomy / Calabi-Yau deep theory).

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§2 Coverage table (Codex vs VHCAG-I)

Cross-referenced against the current Codex corpus. ✓ = covered, △ = partial / different framing, ✗ = not covered. VHCAG-I material maps primarily to 04-algebraic-geometry/09-hodge/, 04-algebraic-geometry/05-surfaces/ + 05-divisors/, 06-riemann-surfaces/04-cohomology/ + 08-vhs/, and the missing complex / Kähler differential-geometric prereq layer in 03-modern-geometry/02-manifolds/ + 05-bundles/ flagged by the KN-II audit.

VHCAG-I topic	Codex unit(s)	Status	Note
Part I (Chs. 1–4) — preliminaries
Complex manifold (atlas, transition functions)	△	△	Touched in 06.01.* (one-variable complex) and 06.07.01 (several variables) without a manifold-level anchor outside the RS chapter. KN-II audit punch-list item 03.02.10 (Complex manifold + Dolbeault complex) shared.
Holomorphic vector bundle	—	✗	Gap (shared with KN-II audit, punch-list 03.05.19). Cited without anchor in 04.09.02-kodaira-vanishing.md and 06.04.02-cech-cohomology-line-bundles.md.
Hermitian metric, Chern connection	—	✗	Gap (shared with KN-II punch-list 03.05.20). Load-bearing for the curvature formulas in Ch. 9 + Ch. 4.
Dolbeault complex $({\Omega }^{p,\bullet },\bar{\partial })$ and Dolbeault cohomology	△ (curve case only)	△	06.04.05-dbar-hilbert-pde.md covers the $\bar{\partial }$ operator and its Hilbert-space PDE setup on a Riemann surface. The general $(p,q)$ Dolbeault complex on a complex manifold has no anchor (KN-II punch-list 03.02.10).
Sheaf cohomology + Dolbeault isomorphism $H^q(X;{\Omega }^p)\cong H_{\bar{\partial }}^{p,q}$	△	△	04.03.01-sheaf-cohomology.md covers sheaf cohomology; the Dolbeault isomorphism specifically is stated in 04.09.01-hodge-decomposition.md Step 4 of the proof outline without a dedicated unit.
Chern classes via curvature	△	△	03.06.06-chern-weil-homomorphism.md + 03.06.04-pontryagin-chern-classes.md cover the construction; KN-II audit punch-list 03.05.20 + 03.06.07 closes remaining gaps.
Part II (Chs. 5–7) — harmonic theory + Hodge package
Ch. 5 — PDE foundation: $d^{\ast }$, $\Delta$, ellipticity, Hodge theorem (real)	△	△	06.04.05-dbar-hilbert-pde.md (Riemann surface) gives the Hilbert-space PDE for $\bar{\partial }$ specifically with lean_mathlib_gap notes for the elliptic-regularity layer; no anchor for the general compact Riemannian Hodge theorem. Gap (high priority). Cited in 04.09.01-hodge-decomposition.md Step 1 without a dedicated unit.
Compact Kähler manifold (Kähler form, $d\omega =0$, equivalent definitions)	—	✗	Gap (shared with KN-II punch-list 03.02.11). This is the central object of the entire volume. Cited in five shipped units (Hodge decomposition, Kodaira vanishing, Hodge index, Hodge decomposition on curves, Newlander-Nirenberg) without a dedicated unit.
Ch. 6 — Kähler identities + Hodge decomposition (complex case)	04.09.01-hodge-decomposition.md	△	A hodge-decomposition unit ships covering the statement of the Hodge decomposition, the Hodge diamond, the symmetries (H1–H4), examples (${\mathbb{P}}^n$, curves, K3, quintic), the construction via harmonic forms, and a sketch of the proof. The Kähler identities themselves are stated in the Master tier "Step 2 — Kähler identities" but are not proved in detail and have no dedicated unit; the $\partial \bar{\partial }$-lemma is not mentioned. KN-II audit punch-list 03.02.12 (Kähler identities + Kähler Hodge decomposition) is shared.
$\partial \bar{\partial }$-lemma	—	✗	Gap (high priority). Load-bearing for formality of compact Kähler manifolds, for the Lefschetz $(1,1)$-theorem proof, and for the cohomological framework on Kähler manifolds. KN-II audit also flagged this.
Dolbeault Laplacian ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$ as self-adjoint elliptic operator	△	△	Mentioned in 04.09.01 Master tier; the operator-theoretic setup is partial in 06.04.05-dbar-hilbert-pde.md for the $L$-valued $(0,q)$-forms on a Riemann surface. Gap for the general complex-manifold case.
Hodge theorem (Dolbeault version): $H^q(X;{\Omega }^p)\cong {\mathcal{H}}^{p,q}(X)$	△	△	Stated as Step 4 in 04.09.01 proof outline. Gap for a dedicated theorem-unit.
Ch. 7 — Lefschetz decomposition + hard Lefschetz
Lefschetz operator $L=\omega \wedge$ and adjoint $\Lambda$	—	✗	Gap. Cited in 04.09.01 Advanced-results section and in 04.05.09-hodge-index-theorem.md Master tier without a dedicated unit.
${\mathfrak{sl}}_2$-action $(L,\Lambda ,H)$ on $H^{\ast }(X;\mathbb{C})$	—	✗	Gap. The structural backbone of the Lefschetz decomposition.
Primitive cohomology $P^k=\ker \Lambda \cap H^k$	—	✗	Gap.
Lefschetz decomposition $H^k={⨁}_r{L}^r{P}^{k-2r}$	—	✗	Gap.
Hard Lefschetz $L^k:H^{n-k}(X)\stackrel{\sim }{\to }{H}^{n+k}(X)$	△	△	Stated in 04.09.01-hodge-decomposition.md Master tier "Hard Lefschetz theorem" as a one-line summary. No proof; no dedicated unit. Gap (high priority). KN-II audit also flagged this.
Hodge-Riemann bilinear relations	△	△	Stated in 04.09.01 Master + 04.05.09 (Hodge index Master). No dedicated unit; no proof. Gap (high priority — load-bearing for hard Lefschetz and Hodge index).
Hodge index theorem on a smooth projective surface	04.05.09-hodge-index-theorem.md	✓	Shipped with both algebraic-geometric (Hartshorne V.1.9) and cohomological (Voisin Ch. 6) framings; master anchor cites VHCAG-I §6.3 with TODO_REF. Single-paragraph weaving + TODO_REF resolution needed.
Part III (Chs. 8–11) — applications and structure theory
Lefschetz $(1,1)$-theorem on smooth projective $X$	△	△	Cited in 04.05.09-hodge-index-theorem.md Master tier as load-bearing for the cohomological Hodge index theorem (identifying $\mathrm{NS}(X)\otimes \mathbb{R}$ with rational Hodge classes in $H^{1,1}$). No dedicated unit. Gap.
Kodaira vanishing $H^q(X;K_X\otimes L)=0$ for $L$ ample, $q>0$	04.09.02-kodaira-vanishing.md	✓	Shipped; master anchor Kodaira 1953 + Voisin Vol I §9 cited. The Voisin-style Akizuki-Nakano-Bochner route to the proof (which sits inside Ch. 9 of VHCAG-I) is mentioned but not at full proof granularity.
Akizuki-Nakano vanishing $H^q(X;{\Omega }^p\otimes L)=0$ for $p+q>n$, $L$ ample	△	△	Mentioned in 04.09.02-kodaira-vanishing.md Intermediate tier as a generalisation. No dedicated unit; Gap (Master-tier deepening).
Kodaira embedding theorem (Kähler $X$ with integral Kähler class ⇒ projective)	—	✗	Gap (high priority). Bridge from compact Kähler manifolds to smooth projective varieties; cited in essentially every algebraic-geometry text without a Codex anchor. Originator citation: Kodaira 1954 Annals.
Hodge structures of weight $k$ (definition, examples, polarisation)	△	△	04.09.01-hodge-decomposition.md Exercise 7 (hard) walks through the definition of a polarised Hodge structure with examples; this is the only treatment. Gap for a dedicated unit (load-bearing for Vol II — see §5).
Variation of Hodge structure	△	△	06.08.01-gauss-manin.md, 06.08.02-vhs-jacobian.md cover the curve case. Gap for a general-base unit; deferred to Vol II audit per §5.
Curve case (one-dimensional Kähler)
Hodge decomposition on a compact Riemann surface	06.04.03-hodge-decomposition-curves.md	✓	Shipped; master anchor "Hodge 1941 + Griffiths-Harris §0.6–§0.7 + Voisin Vol I §5–§6" cited with TODO_REF. Excellent coverage of the one-dim case.
Riemann's bilinear relations	△	△	06.06.07-riemann-bilinear.md covers the period-matrix bilinear relations on a Riemann surface (the curve specialisation of the Hodge-Riemann bilinear relations); ships with Riemann 1857 + Voisin citations.
$\bar{\partial }$ Hilbert-space PDE (one-dim)	06.04.05-dbar-hilbert-pde.md	✓	Shipped with master anchor "Hörmander 1965 + Andreotti-Vesentini 1965 + Hodge 1941 + Demailly + Voisin §5"; covers the one-variable PDE foundation that VHCAG-I Ch. 5 lays out in general.

Aggregate coverage estimate: ~30–35% of VHCAG-I has corresponding Codex units. Part I (preliminaries) is ~30% covered: sheaf cohomology + Chern-Weil + Chern classes are shipped; complex manifold, holomorphic vector bundle, Hermitian metric / Chern connection, and the general Dolbeault complex are all gaps shared with the KN-II audit punch-list. Part II (harmonic theory + Hodge package) is ~25% covered: the statement of the Hodge decomposition ships in 04.09.01-hodge-decomposition.md with examples and a proof outline; the PDE input (Ch. 5), the Kähler identities (Ch. 6), the $\partial \bar{\partial }$-lemma (Ch. 6), the Lefschetz decomposition and hard Lefschetz and Hodge-Riemann bilinear relations (Ch. 7) are all gaps. The curve case is ~80% covered by the 06.04.03 + 06.04.05 + 06.06.07 cluster — this is the load-bearing in-Codex evidence that the Hodge programme on Riemann surfaces is well-developed, and the gap is for the general-dimensional Kähler case. Part III (applications + structure theory) is ~40% covered: Kodaira vanishing ships; the Hodge index theorem ships; the Lefschetz $(1,1)$-theorem, Akizuki-Nakano, Kodaira embedding, and the general theory of Hodge structures and VHS are gaps.

Silent VHCAG-I dependencies in the Codex. Five shipped units cite VHCAG-I as a master anchor; in each case the load-bearing structural fact from Vol I is invoked but not anchored:

1. 04.09.01-hodge-decomposition.md — Master anchor cites Voisin Vol I; cites the Kähler identities, the $\partial \bar{\partial }$-lemma (not by name, but implicitly via the harmonic-form construction), and hard Lefschetz + Hodge-Riemann in the Advanced-results section. None of these have dedicated units. references[].source: TODO_REF for the Voisin citation.
2. 04.09.02-kodaira-vanishing.md — Master anchor cites Voisin Vol I §9; the Akizuki-Nakano-Bochner identity is the load-bearing PDE input to the Kodaira-vanishing proof and is mentioned only in passing. references[].source: TODO_REF for the Voisin citation.
3. 04.05.09-hodge-index-theorem.md — Master anchor explicitly cites "Voisin Hodge Theory and Complex Algebraic Geometry I §6.3" with TODO_REF; the Hodge-Riemann bilinear relations and the Lefschetz $(1,1)$-theorem are the load-bearing inputs to the cohomological Hodge index theorem and have no anchor.
4. 06.04.03-hodge-decomposition-curves.md — Master anchor cites Voisin Vol I §5–§6 with TODO_REF. The curve case is well-covered in-Codex but the upstream general-dim Kähler-package framing is the gap.
5. 06.04.05-dbar-hilbert-pde.md — Master anchor cites Voisin Vol I §5 with TODO_REF. The curve case of the $\bar{\partial }$ PDE is shipped; the general Kähler-manifold version (VHCAG-I Ch. 5 + Ch. 9 for $L$-valued forms) is the gap.

Net effect: the existing Codex 04-algebraic-geometry/09-hodge/ chapter covers ~50% of the relevant material by stating Hodge decomposition + Kodaira vanishing + Hodge index in self-contained units; the proof-route machinery (PDE foundation, Kähler identities, $\partial \bar{\partial }$-lemma, Lefschetz decomposition, hard Lefschetz, Hodge-Riemann) is essentially absent as dedicated units and is only sketched inside the existing statement-units. The 06-riemann-surfaces/04-cohomology/ chapter develops the curve case rigorously; the gap is the lift from curves to general compact Kähler manifolds. The VHCAG-I audit is therefore load-bearing across two chapters — 04-algebraic-geometry/09-hodge/ (new Kähler-package units) and 03-modern-geometry/02-manifolds/ / 05-bundles/ (the complex / Kähler differential-geometric prereq layer shared with the KN-II punch-list).

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

Priority 0 — prerequisite chain shared with the KN-II audit:

These are listed here because VHCAG-I assumes them; they are already on the KN-II punch-list (FT 3.19, priority 1) and the KN-I punch-list (FT 3.18). Ship once, cite from both audits.

• 03.02.09 Almost-complex structure (manifold-level) (KN-II punch-list priority 1) — load-bearing for VHCAG-I Ch. 2.
• 03.02.10 Complex manifold and the Dolbeault complex (KN-II punch-list priority 1) — load-bearing for VHCAG-I Chs. 2–3 and Ch. 5.
• 03.05.19 Holomorphic vector bundle (KN-II punch-list priority 1) — load-bearing for VHCAG-I Ch. 3 + Ch. 9 ($L$-valued forms in Kodaira vanishing).
• 03.05.20 Hermitian metric on a complex bundle; Chern connection (KN-II punch-list priority 1) — load-bearing for VHCAG-I Ch. 4 + Ch. 9.
• 03.02.11 Hermitian manifold and the Kähler form (KN-II punch-list priority 1) — load-bearing for VHCAG-I Ch. 6 onwards. This is the central object of VHCAG-I; without it no Vol-I material beyond Part I can be properly stated.
• 03.02.12 Kähler identities and the Hodge decomposition (Kähler version) (KN-II punch-list priority 1) — load-bearing for VHCAG-I Ch. 6 + Ch. 7. The KN-II punch-list ships this as a single unit; the Voisin audit recommends splitting it into two for full Vol-I coverage (see priority-1 below).

Priority 1 — Part II / Kähler-package core (PDE foundations + Hodge decomposition proof + Lefschetz decomposition):

1. 04.09.03 PDE foundation: harmonic theory on a compact Riemannian manifold. The Hodge theorem in its real form: on a compact Riemannian $(X,g)$, the Laplacian $\Delta =d{d}^{\ast }+d^{\ast }d$ is a self-adjoint elliptic operator, $\ker \Delta ={\mathcal{H}}^k(X)$ is finite-dimensional, and $\Omega^k(X) = \mathcal{H}^k \oplus d\Omega^{k-1} \oplus d^*\Omega^{k+1}$ (the Hodge orthogonal decomposition). Proof via Gårding's inequality, elliptic regularity, and Fredholm theory of self-adjoint elliptic operators on compact manifolds. VHCAG-I §5 anchor; Wells §IV secondary; Demailly Ch. VI secondary. ~2000 words. Master-tier dominant; Intermediate gives the statement and the use of harmonic representatives; Beginner gives the slogan "every cohomology class has a unique harmonic representative". Closes the silent gap behind 04.09.01-hodge-decomposition.md Step 1 + 06.04.05-dbar-hilbert-pde.md (lifts the curve case to the general case). Originator-prose citation: W. V. D. Hodge, The Theory and Applications of Harmonic Integrals (Cambridge 1941; rigorous PDE proof Weyl 1943, simplified Kodaira-Spencer 1953).
2. 04.09.04 Kähler identities and the Hodge decomposition (general compact Kähler manifold; the proof). This is the Voisin Ch. 6 centrepiece. The Kähler identities $[\Lambda ,\bar{\partial }]=-i{\partial }^{\ast }$, $[\Lambda ,\partial ]=i{\bar{\partial }}^{\ast }$; the consequence $\Delta_d = 2\Delta_\partial = 2\Delta_{\bar\partial}$ on a Kähler manifold; the proof of the Hodge decomposition $H^k(X; \mathbb{C}) = \bigoplus_{p + q = k} H^{p, q}(X)$ on a compact Kähler manifold. The Hodge symmetry $\overline{H^{p,q}}=H^{q,p}$ as a corollary. VHCAG-I §6 anchor; Griffiths-Harris §0.6–§0.7 secondary; Huybrechts §3.2 secondary; Demailly Ch. VI secondary. ~2200 words. Master-tier dominant. Sharpens 04.09.01-hodge-decomposition.md with the full differential-geometric proof. Shared with KN-II punch-list 03.02.12; ship as a Hodge-chapter unit referenced from the KN-II Kähler section.
3. 04.09.05 $\partial \bar{\partial }$-lemma. On a compact Kähler manifold, every $d$-exact, $\partial$-closed, $\bar{\partial }$-closed $(p,q)$-form is $\partial \bar{\partial }$-exact: $\alpha =\partial \bar{\partial }\beta$ for some $(p-1,q-1)$-form $\beta$. Consequences: formality of compact Kähler manifolds (Deligne-Griffiths-Morgan-Sullivan 1975, Real homotopy theory of Kähler manifolds); independence of the Hodge filtration from choice of Kähler metric. VHCAG-I §6.1.5 anchor; Griffiths-Harris §0.7 secondary; Huybrechts §3.A secondary; Deligne-Griffiths-Morgan-Sullivan 1975 Invent. Math. 29 originator citation. ~1500 words. Master-tier dominant. Closes the silent gap in 04.09.01-hodge-decomposition.md and is load-bearing for the Lefschetz $(1,1)$-theorem.
4. 04.09.06 Lefschetz operator $L$, dual $\Lambda$, and the ${\mathfrak{sl}}_2$-decomposition. With $\omega$ the Kähler form on a compact Kähler $X$, the Lefschetz operator $L=\omega \wedge$ on ${\Omega }^{\bullet }(X)$, its adjoint $\Lambda =$ contraction with $\omega$, and the counting operator $H=[L,\Lambda ]$ form an ${\mathfrak{sl}}_2$- representation on $H^{\ast }(X;\mathbb{C})$. The primitive cohomology $P^k=\ker \Lambda \cap H^k(X)$ (for $k\le n$) and the Lefschetz decomposition $H^k={⨁}_r{L}^r{P}^{k-2r}$. VHCAG-I §7.1 anchor; Griffiths-Harris §0.7 secondary; Demailly Ch. VI secondary. ~1800 words. Master-tier dominant; Intermediate gives the statement of the ${\mathfrak{sl}}_2$-action and the primitive decomposition.
5. 04.09.07 Hard Lefschetz theorem. On a compact Kähler manifold $X$ of complex dimension $n$ with Kähler form $\omega$, the cup-product map $L^k:H^{n-k}(X;\mathbb{C})\stackrel{\sim }{\to }{H}^{n+k}(X;\mathbb{C})$ is an isomorphism for $0\le k\le n$. Proof via the Kähler identities + the Lefschetz ${\mathfrak{sl}}_2$-decomposition (representation- theoretic argument from 04.09.06). VHCAG-I §7.1.2 anchor; Griffiths-Harris §0.7 secondary; Huybrechts §3.3 secondary. ~1500 words. Master-tier dominant; Intermediate gives the statement. Closes the silent gap in 04.09.01-hodge-decomposition.md Advanced-results section.
6. 04.09.08 Hodge-Riemann bilinear relations. For a compact Kähler manifold of dimension $n$ with Kähler class $\omega$ and primitive cohomology $P^{p,q}=P^k\cap H^{p,q}$ with $p+q=k\le n$: the first bilinear relation $Q(\alpha ,\beta )=0$ for $\alpha \in P^{p,q},\beta \in P^{p^{\prime },q^{\prime }}$ with $(p,q)\ne (q^{\prime },p^{\prime })$; the second bilinear relation $i^{p - q} (-1)^{k(k-1)/2} Q(\alpha, \bar\alpha) > 0$for$0 \neq \alpha \in P^{p, q}$, where $Q(\alpha ,\beta )={\int }_X{\omega }^{n-k}\wedge \alpha \wedge \beta$. Corollaries: the Hodge index theorem (surface case = 04.05.09); polarised Hodge structures of weight $k$. VHCAG-I §7.1.3 + §7.2 anchor; Griffiths-Harris §0.7 secondary; Riemann 1857 Theorie der Abel'schen Functionen originator citation (the period-relations origin); Hodge 1941 + Weil 1958 Variétés Kählériennes originator citations (the higher-dim Kähler synthesis). ~2000 words. Master-tier dominant. Closes the silent gap in 04.05.09-hodge-index-theorem.md Master tier and 04.09.01-hodge-decomposition.md Advanced-results section.

Priority 2 — Part III applications (embedding + vanishing deepening + Lefschetz $(1,1)$):

7. 04.09.09 Lefschetz $(1,1)$-theorem. On a smooth projective variety $X$, the cycle-class map $\mathrm{NS}(X) \otimes \mathbb{Q} \xrightarrow{\sim} H^{1, 1}(X; \mathbb{Q}) \cap H^2(X; \mathbb{Q})$ is an isomorphism (Lefschetz 1924). Bridge from algebraic divisors to Hodge classes; the $(1,1)$ case of the Hodge conjecture, which is the only fully-proved case. VHCAG-I §11.3 anchor; Griffiths-Harris §1.2 secondary; Lefschetz 1924 L'analysis situs et la géométrie algébrique originator citation. ~1500 words. Master-tier dominant. Closes the silent gap in 04.05.09-hodge-index-theorem.md and provides the bridge to Vol II Hodge-conjecture material.
8. 04.09.10 Akizuki-Nakano vanishing theorem. $H^q(X; \Omega^p_X \otimes L) = 0$for$L$ampleonasmoothprojective$X$ of dimension $n$ and $p+q>n$. Generalises Kodaira vanishing (the $p=n$ case); uses the Akizuki-Nakano-Bochner identity on a Kähler manifold. VHCAG-I §9 anchor; Akizuki-Nakano 1954 Proc. Japan Acad. 30 originator citation ("Note on the elliptic character of a sheaf cohomology and proper K-extension"). ~1500 words. Master-tier dominant. Deepens 04.09.02-kodaira-vanishing.md.
9. 04.09.11 Kodaira embedding theorem. A compact Kähler manifold $X$ admits a holomorphic embedding into ${\mathbb{P}}^N$ for some $N$ iff $X$ carries an integral Kähler class — equivalently, iff $X$ carries a positive holomorphic line bundle. Bridge from compact Kähler manifolds to smooth projective varieties. VHCAG-I §10 anchor; Kodaira 1954 Annals of Math. 60 originator citation. ~2000 words. Three-tier. High-leverage; ties together the Kähler-manifold theory with smooth projective varieties.

Priority 3 — Hodge structures formalism (foundation for Vol II audit):

10. 04.09.12 Hodge structure of weight $k$ and polarised Hodge structure. Definition: a $\mathbb{Q}$-Hodge structure of weight $k$ on a finite-dim $\mathbb{Q}$-vector space $V$ is a decomposition $V_{\mathbb{C}}={⨁}_{p+q=k}{V}^{p,q}$ with $\overline{V^{p,q}}=V^{q,p}$. Equivalently, a decreasing filtration $F^{\bullet }$ on $V_{\mathbb{C}}$ with $V^{p,q}=F^p\cap \overline{F^q}$. Polarisation: a $\mathbb{Q}$-bilinear form $Q : V \times V \to \mathbb{Q}$ satisfying the Hodge-Riemann bilinear relations. The cohomology of a compact Kähler manifold is the motivating example; abstract Hodge structures abstract this. VHCAG-I §11 anchor; Griffiths-Harris §3.1 secondary; Deligne 1971–72 originator citations (the mixed Hodge case, deferred to Vol II audit). ~1800 words. Three-tier. Sets up the Vol II audit.

Priority 4 — survey pointers (Master-only, weaving edits to existing units):

11. Pointer in 04.09.01-hodge-decomposition.md to the new 04.09.03 (PDE foundation), 04.09.04 (Kähler identities + Hodge decomposition proof), 04.09.05 ($\partial \bar{\partial }$-lemma), and resolve the Voisin TODO_REF to VHCAG-I Ch. 6 with concrete section locators. Single-paragraph weaving edit per linked unit; resolves the audit-flagged TODO_REF.
12. Pointer in 04.09.02-kodaira-vanishing.md to 04.09.10 (Akizuki-Nakano) as the natural generalisation, to the new 03.05.20 (Chern connection) as the prerequisite layer, and to 04.09.11 (Kodaira embedding) as the application-direction companion. Resolve the Voisin TODO_REF to VHCAG-I Ch. 9.
13. Pointer in 04.05.09-hodge-index-theorem.md to 04.09.08 (Hodge-Riemann) and 04.09.09 (Lefschetz $(1,1)$) as the load-bearing structural inputs. Resolve the Voisin TODO_REF to VHCAG-I §6.3.
14. Pointer in 06.04.03-hodge-decomposition-curves.md to 04.09.04 (general Kähler Hodge decomposition) as the upstream framing of which the curve case is the dim-1 specialisation. Resolve the Voisin TODO_REF to VHCAG-I §5–§6.
15. Pointer in 06.04.05-dbar-hilbert-pde.md to 04.09.03 (general PDE foundation) and the new 03.02.10 (Dolbeault complex). Resolve the Voisin TODO_REF to VHCAG-I §5.
16. Pointer in 04.09.07 (new hard Lefschetz unit) to the existing 04.05.09-hodge-index-theorem.md as the surface specialisation application of the Hodge-Riemann positivity input.
17. Pointer in 04.10.01-moduli-of-curves.md and 06.08.* (VHS chapter on Riemann surfaces) to 04.09.12 (Hodge structures) as the abstract framework. Single-paragraph weaving.

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§4 Implementation sketch (P3 → P4)

For a full VHCAG-I coverage pass, priority-0 + priority-1 are the minimum set and close the Kähler-package proof-route gaps that the existing Codex statement-units silently depend on. Realistic production estimate (mirroring earlier KN-I / KN-II / Milnor / Bott-Tu batches):

• ~3.5–4 hours per unit. VHCAG-I units skew above the corpus average because the Master tier requires (i) the PDE technical machinery from Ch. 5, (ii) the Kähler-identities computation from Ch. 6, and (iii) the ${\mathfrak{sl}}_2$-representation-theoretic bookkeeping from Ch. 7 — all of which are not casual material.
• Priority 0: 6 units shared with the KN-II + KN-I punch-lists (do not double-count); flagged here for prerequisite tracking only. Already in those audits' production estimates (~21 hours in the KN-II audit).
• Priority 1: 6 units × ~4 hours = ~24 hours. Heart of the audit; closes the Kähler-package proof-route gaps across the algebraic-geometry Hodge chapter and supplies the upstream framing for the Riemann-surface case.
• Priority 2: 3 units × ~3.5 hours = ~10.5 hours. Closes the Lefschetz $(1,1)$, Akizuki-Nakano, and Kodaira embedding gaps; brings Part III to FT-equivalence.
• Priority 3: 1 unit × ~3.5 hours = ~3.5 hours. Sets up the Vol II audit.
• Priority 4: 7 weaving edits × ~30 min = ~3.5 hours. Resolves five TODO_REF blocks and connects new units to existing units.
• Total for full coverage: ~42 hours, roughly a focused 6–8 day window. Priority-0 (carried by the KN-II audit) + priority-1 alone is ~24 hours and closes the load-bearing Kähler-package proof-route gaps; with priority-2 (~10.5 hours more) Part III reaches FT-equivalence.

Originator-prose targets. Per docs/plans/FASTTRACK_EQUIVALENCE_PLAN.md §10, the following units should carry originator-prose citations:

• W. V. D. Hodge, The Theory and Applications of Harmonic Integrals (Cambridge University Press 1941; 2nd ed. 1952) — originating monograph for the harmonic-form approach to cohomology of complex manifolds and for the Hodge decomposition itself. Cite in 04.09.03 (PDE foundation — the Hodge theorem in real form), 04.09.04 (Kähler Hodge decomposition), and confirm in the existing 04.09.01-hodge-decomposition.md (Historical context already cites it).
• K. Kodaira, "On a differential-geometric method in the theory of analytic stacks," Proc. Nat. Acad. Sci. USA 39 (1953) 1268–1273 — originating paper for the Kodaira vanishing theorem. Already cited in 04.09.02-kodaira-vanishing.md.
• K. Kodaira, "On Kähler varieties of restricted type," Annals of Math. 60 (1954) 28–48 — originating paper for the Kodaira embedding theorem. Cite in 04.09.11 (new Kodaira embedding unit).
• A. Weil, Introduction à l'étude des variétés kählériennes (Actualités Sci. Ind. 1267, Hermann, Paris 1958) — the foundational modern systematic monograph on Kähler manifolds, synthesising the Hodge / Chern / Kodaira programme. Cite in 04.09.04 (Kähler identities
  • Hodge decomposition) and 04.09.08 (Hodge-Riemann bilinear relations).
• Y. Akizuki, S. Nakano, "Note on the elliptic character of a sheaf cohomology and proper K-extension," Proc. Japan Acad. 30 (1954) 266–272 — originating paper for the Akizuki-Nakano vanishing theorem. Cite in 04.09.10.
• S. Lefschetz, L'analysis situs et la géométrie algébrique (Gauthier-Villars, Paris 1924) — originating monograph for the Lefschetz $(1,1)$-theorem and the original (topological / monodromy) proof of hard Lefschetz. Cite in 04.09.07 (hard Lefschetz) and 04.09.09 (Lefschetz $(1,1)$).
• P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, "Real homotopy theory of Kähler manifolds," Invent. Math. 29 (1975) 245–274 — originating paper for formality of compact Kähler manifolds via the $\partial \bar{\partial }$-lemma. Cite in 04.09.05 as the canonical consequence-of-$\partial \bar{\partial }$-lemma reference.
• C. Voisin, Hodge Theory and Complex Algebraic Geometry I (Cambridge SAM 76, CUP 2002) — the book itself, as the definitive modern systematic English-language treatment. Cite throughout priority-1 + priority-2 units as the master anchor.
• Forward pointer to Vol II: P. Deligne, "Théorie de Hodge I, II, III" (ICM 1970; Publ. Math. IHES 40 (1971) 5–57; 44 (1974) 5–77) — for the mixed Hodge structures programme. Cited as a pointer-only in the new 04.09.12 (Hodge structures) unit's "see Voisin Vol II" closing note.

Notation crosswalk. VHCAG-I uses $L$ for the Lefschetz operator and $\Lambda$ for its adjoint (the contraction-with-$\omega$ operator); the Codex 04.09.01-hodge-decomposition.md already uses these conventions. VHCAG-I writes $H^{p,q}(X)$ for the Hodge components and ${\mathcal{H}}^{p,q}$ for harmonic-form representatives; the Codex follows. VHCAG-I uses ${\Omega }^p$ ambiguously for both the sheaf of holomorphic $p$-forms and smooth $(p,0)$-forms; the Codex unit specs (per docs/specs/UNIT_SPEC.md §11) should consistently write ${\Omega }_X^p$ for the sheaf and ${\mathcal{A}}^{p,q}$ for the sheaf of smooth $(p,q)$-forms. VHCAG-I's $P^k$ for primitive cohomology is standard. Record these in §Notation paragraphs of 04.09.04, 04.09.06, and 04.09.08.

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§5 What this plan does NOT cover

• Voisin Vol II (Hodge Theory and Complex Algebraic Geometry, II, Cambridge SAM 77, CUP 2003) is deferred to its own audit (Fast Track 3.27b — not yet on the FT booklist; recommend adding as a sibling entry). Vol II covers: mixed Hodge structures (Deligne 1971–74), the Hodge conjecture, variations of Hodge structure and the period map (Griffiths 1968–72; Schmid 1973), Noether-Lefschetz theory, algebraic cycles and the Bloch-Beilinson conjectures, normal functions. Vol I priority-3 unit 04.09.12 (Hodge structures) is the only unit in this audit that touches Vol-II-flavoured material, and it does so as setup only.
• Calabi-Yau / Kähler-Einstein deep theory. The Calabi conjecture (Yau 1977; Comm. Pure Appl. Math. 31, 339–411), Aubin's work on the negative-curvature case, Tian's stability programme, and the Donaldson-Tian-Yau / K-stability theory of Kähler-Einstein metrics on Fano manifolds are deferred to a dedicated future audit. VHCAG-I itself defers this material to a pointer; the Codex follows. A future Joyce — Compact Manifolds with Special Holonomy audit or Tian — Canonical Metrics in Kähler Geometry audit is the natural home for the Calabi-Yau / Kähler-Einstein punch-list.
• The four-manifold / Donaldson / Seiberg-Witten programme. Connects to VHCAG-I's Kähler-package via the Kähler-surface special case but is a separate research programme; deferred to a dedicated future audit (Donaldson-Kronheimer, The Geometry of Four-Manifolds, OUP 1990; or Salamon, Spin Geometry and Seiberg-Witten Invariants, ETH 2000).
• A line-number-level inventory of every named theorem in VHCAG-I (full P1 audit; deferred — VHCAG-I is 322 pp. with hundreds of named results). The production pass should do this when a local PDF is on disk.
• Exercise-pack production. VHCAG-I has substantial exercise sections; the Codex exercise-pack production is P3-priority-3 follow-up after the priority-1 + priority-2 units ship.
• Saito's mixed Hodge modules and the derived-category Hodge-theoretic framework. Deferred to the Vol II audit + a possible dedicated Saito-mixed-Hodge-modules audit.
• $p$-adic Hodge theory (Fontaine, Faltings, Tsuji, Scholze) — this is a different programme entirely and is not in scope for the VHCAG-I audit.

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§6 Acceptance criteria for FT equivalence (VHCAG-I)

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

• The KN-II priority-1 units have shipped (strict prereq — the Kähler-manifold definition and the complex-bundle / Hermitian / Chern- connection layer must exist).
• ≥95% of VHCAG-I's named theorems in Chs. 5–10 map to Codex units (currently ~25%; after priority-1 units this rises to ~75%; after priority-1 + priority-2 to ~90%; full ≥95% requires priority-3 + selective priority-4).
• ≥90% of VHCAG-I's worked computations in Chs. 5–10 have a direct unit or are referenced from a unit that covers them.
• The five existing TODO_REF blocks citing VHCAG-I (in 04.09.01, 04.09.02, 04.05.09, 06.04.03, 06.04.05) are resolved with concrete section locators.
• Notation decisions are recorded (see §4).
• Pass-W weaving connects the new units to 04-algebraic-geometry/09-hodge/, 06-riemann-surfaces/04-cohomology/, and 04-algebraic-geometry/05-surfaces/ via lateral connections.

The 6 priority-1 units close most of the equivalence gap given the KN-II prereqs are in place. Priority-2 closes Part III (Lefschetz $(1,1)$

• Akizuki-Nakano + Kodaira embedding). Priority-3 sets up the Vol II audit. Priority-4 are weaving edits.

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§7 Sourcing

• Local copy. Not available in reference/textbooks-extra/ or reference/fasttrack-texts/03-modern-geometry/ as of 2026-05-18. VHCAG-I is under active CUP / SMF copyright and was not retrieved within the time budget. Recommend obtaining a copy (BUY per FT booklist entry 3.27) before the production pass; add to reference/textbooks-extra/ as Voisin-HodgeTheoryAndComplexAlgebraicGeometry-Volume-I.pdf when available.
• Translation note. The English edition (Schneps translation, CUP 2002) is the canonical Codex reference; the French original (Théorie de Hodge et géométrie algébrique complexe, Cours Spécialisés 10, SMF 2002) is contemporaneous and content-equivalent. Cite the English edition by default.
• License. Under active CUP copyright. For educational use cite as C. Voisin, Hodge Theory and Complex Algebraic Geometry I (translated by L. Schneps), Cambridge Studies in Advanced Mathematics 76, Cambridge University Press 2002.
• Peer-source fallbacks for the production pass when the local PDF is delayed:
  • D. Huybrechts, Complex Geometry: An Introduction (Universitext, Springer 2005) — most accessible parallel; Chs. 2–5 cover essentially the same material at a slightly less dense pace.
  • R. O. Wells, Differential Analysis on Complex Manifolds (GTM 65, Springer 3rd ed. 2008, with appendix by O. García-Prada) — classical hard-analysis parallel; Chs. I–V.
  • P. Griffiths, J. Harris, Principles of Algebraic Geometry (Wiley 1978; reprinted Wiley Classics 1994) — classical algebraic-geometry- flavoured parallel; Chs. 0–1.
  • J.-P. Demailly, Complex Analytic and Differential Geometry (OpenContent monograph, current online edition at https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf) — encyclopedic PDE-rigorous parallel; Chs. VI–X. Free. Strongest free fallback for the harmonic-theoretic / PDE / Bochner-Kodaira- Nakano material.

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Audit mode flag. This plan was produced in REDUCED mode (no local PDF). Production-pass agents should treat the priority-1 + priority-2 section locators (VHCAG-I §5, §6, §7.1.2, §7.1.3, §9, §10, §11.3) as canonical-but-unverified and confirm against the local PDF when available. The TOC-level structure used here is well-attested across Huybrechts, Wells, Griffiths-Harris, and Demailly, but exact section numbering within VHCAG-I should be verified at production time.