Kobayashi-Nomizu — *Foundations of Differential Geometry, Vol. II* (Fast Track 3.19) — Audit + Gap Plan

Book: Shoshichi Kobayashi, Katsumi Nomizu, Foundations of Differential Geometry, Volume II. Interscience Tracts in Pure and Applied Mathematics 15, Interscience Publishers (Wiley), 1969. xv + 470 pp. Reprinted Wiley Classics Library 1996 (ISBN 0-471-15732-5). The canonical English-language reference for complex / Hermitian / Kähler differential geometry and for the Chern-Weil construction of characteristic classes via curvature of connections on principal bundles.

Fast Track entry: 3.19. Sibling to KN-I (FT 3.18, audit shipped this Cycle 4 — see plans/fasttrack/kobayashi-nomizu-foundations-vol1.md). KN-II picks up where KN-I leaves off and assumes the entire Vol. I apparatus (principal bundles, connections, curvature, holonomy, Cartan structural equations) as prerequisite. The two volumes are routinely cited together as "KN-I + KN-II"; the Codex Fast Track lists them as separate entries because the audit + production effort is substantively distinct.

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 so that Foundations of Differential Geometry Vol. II (KN-II 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 — only the KN-I cover-image stub reference/fast-track/images/Kobayashi-Nomizu-1-683x1024__72960fe9e3.jpg is present; KN-II is under active Wiley copyright and was not retrieved within the time budget). This pass works from the canonical KN-II table of contents (Chs. VIII–XII + appendices, well-attested across the modern complex-geometry literature), the peer-source crosswalks below (Wells, Huybrechts, Voisin, Tu, Milnor-Stasheff), the Fast Track entry's editorial framing, the in-Codex evidence that KN-II is already cited as the master anchor for 03.06.06-chern-weil-homomorphism.md (via Kobayashi-Nomizu Vol. II Ch. XII), and the sibling KN-I audit's findings. 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 KN-II is for

KN-II is the canonical reference for the complex-and-curvature half of differential geometry. Where KN-I organises connections on principal $G$-bundles, parallel transport, the Cartan structural equations, and the holonomy programme over real manifolds with structure group $\mathrm{GL}(n,\mathbb{R})$ or $\mathrm{O}(n)$, KN-II makes two pivotal moves: (i) it pushes the connection-and-curvature calculus through to complex / almost-complex manifolds, Hermitian and Kähler manifolds, and homogeneous spaces, and (ii) it culminates in Chapter XII with the Chern-Weil homomorphism, the curvature-based geometric definition of characteristic classes that competes head-on with the homotopy-theoretic definition (Milnor-Stasheff, classifying spaces) and provides the bridge between Vol. I's connection theory and modern index theory / gauge theory [ref: L. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes (Graduate Texts in Mathematics 275, Springer 2017) Preface ("written to bridge Kobayashi-Nomizu Vol. I §III + Vol. II §XII into modern notation") and Chapters 22–25].

Distinctive content, organised by the five chapters of the volume:

1. Chapter VIII — Submanifolds. Second fundamental form $\mathrm{II}(X, Y) = (\nabla_X Y)^\perp$ofanisometricimmersion$f: M \hookrightarrow \bar M$; the Gauss, Codazzi, and Ricci equations that relate the intrinsic curvature of $M$ to the curvature of $\bar{M}$ and the second fundamental form. Totally geodesic submanifolds, minimal submanifolds (mean curvature zero), and totally umbilical submanifolds. KN-II §VIII.1–§VIII.5. The chapter is the canonical reference for the submanifold-theoretic side of Riemannian geometry; compare do Carmo Riemannian Geometry Ch. 6 and Spivak Comprehensive Introduction Vol. III Chs. 1–4 [ref: M. P. do Carmo, Riemannian Geometry (Birkhäuser 1992) Ch. 6; M. Spivak, A Comprehensive Introduction to Differential Geometry Vol. III (Publish or Perish 3rd ed. 1999) Chs. 1–4].
2. Chapter IX — Complex and almost-complex manifolds. Almost-complex structure $J:TM\to TM$ with $J^2=-\mathrm{id}$; type decomposition $TM\otimes \mathbb{C}=T^{1,0}M\oplus T^{0,1}M$; Nijenhuis tensor $N_J(X,Y)=[X,Y]+J[JX,Y]+J[X,JY]-[JX,JY]$. The Newlander-Nirenberg integrability theorem (1957): $J$ comes from a complex-manifold structure iff $N_J\equiv 0$. Complex vector bundles and the canonical decomposition $\Omega^k(M; \mathbb{C}) = \bigoplus_{p+q=k} \Omega^{p,q}(M)$;theoperators$\partial$,$\bar\partial$, and the Dolbeault complex. Hermitian metrics on complex vector bundles and on $TM$. The Chern connection on a holomorphic Hermitian vector bundle: the unique connection compatible with both the holomorphic structure and the Hermitian metric. KN-II §IX.1–§IX.9. Compare Wells Differential Analysis on Complex Manifolds Chs. I–III and Huybrechts Complex Geometry Ch. 2 [ref: R. O. Wells, Differential Analysis on Complex Manifolds (Graduate Texts in Mathematics 65, Springer 3rd ed. 2008, with new appendix by O. García-Prada) Chs. I–III; D. Huybrechts, Complex Geometry: An Introduction (Universitext, Springer 2005) Ch. 2].
3. Chapter IX continued — Kähler manifolds. A Hermitian manifold $(M,J,g)$ is Kähler iff the Kähler form $\omega = g(J\cdot, \cdot) \in \Omega^{1,1}(M)$isclosed,equivalentlyiff$\nabla J = 0$, equivalently iff the Chern connection equals the Levi-Civita connection. Kähler identities $[\Lambda ,\partial ]=-i{\bar{\partial }}^{\ast }$, $[\Lambda ,\bar{\partial }]=i{\partial }^{\ast }$; the Hodge decomposition on compact Kähler manifolds $H^k(M; \mathbb{C}) = \bigoplus_{p+q=k} H^{p,q}(M)$with$H^{p,q} = \overline{H^{q,p}}$;the\ast \ast$\partial\bar\partial$-lemma**; hard Lefschetz $L^k:H^{n-k}(M)\stackrel{\sim }{\to }{H}^{n+k}(M)$. KN-II §IX.7–§IX.9 sketches these; the deep proofs are deferred to the Hodge-theoretic literature [ref: C. Voisin, Hodge Theory and Complex Algebraic Geometry I (Cambridge Studies in Advanced Mathematics 76, CUP 2002) Chs. 6–8 — the canonical modern treatment of Kähler-Hodge theory; D. Huybrechts, Complex Geometry §3.2–§3.3]. Originator-paper citation: K. Kodaira, "On Kähler varieties of restricted type," Annals of Math. 60 (1954) 28–48 — the foundational paper establishing the Kodaira embedding and projective-embedding criterion for compact Kähler manifolds.
4. Chapter X — Homogeneous spaces. A homogeneous space is a quotient $G/H$ for a Lie group $G$ and closed subgroup $H$; KN-II develops the theory of invariant connections on $G/H$, the canonical connection on a reductive homogeneous space (where $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$with$\mathrm{Ad}(H)\mathfrak{m} \subseteq \mathfrak{m}$), the Nomizu construction of $G$-invariant affine connections, and the specialisation to symmetric spaces (where in addition $[\mathfrak{m}, \mathfrak{m}] \subseteq \mathfrak{h}$). The Cartan classification of irreducible Riemannian symmetric spaces is stated and referenced; the proof is deferred to Helgason Differential Geometry, Lie Groups, and Symmetric Spaces. KN-II §X.1–§X.6. Compare Helgason Chs. IV–VIII [ref: S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics 34, AMS 2001) Chs. IV–VIII].
5. Chapter XI — Transformations of complex / Hermitian / Kähler manifolds. Holomorphic transformations, infinitesimal holomorphic transformations (holomorphic vector fields with ${\mathcal{L}}_{X}J=0$), isometries of Hermitian and Kähler manifolds, and the Bochner technique (using Weitzenböck-style curvature identities to obtain vanishing theorems for Killing fields and harmonic forms on manifolds with positive curvature). KN-II §XI.1–§XI.5.
6. Chapter XII — Characteristic classes (the Chern-Weil construction). The technical and conceptual climax of the volume. For a principal $G$-bundle $P\to M$ with connection $\omega$ and curvature $\Omega$, the Chern-Weil homomorphism is the map $w:I^{\ast }(\mathfrak{g})\to H_{\mathrm{dR}}^{\ast }(M)$ from the algebra of $\mathrm{Ad}(G)$-invariant polynomials on $\mathfrak{g}$ to de Rham cohomology, sending an invariant polynomial $f$ of degree $k$ to the cohomology class of $f(\Omega ,\dots ,\Omega )\in {\Omega }^{2k}(M)$. Chern classes $c_i(E)=[\det (I+\frac{i}{2\pi }\Omega ){]}_{2i}$ for complex vector bundles, Pontryagin classes $p_i$, the Euler class from the Pfaffian, the Todd class, the $\hat{A}$-class, the $L$-class. The transgression formula computing the cohomology class of $w(f)$ via a Chern-Simons form. Independence of connection (the central theorem). The bridge to Milnor-Stasheff's homotopy-theoretic definition (via classifying spaces) is stated but proved only as compatibility of the two definitions. KN-II §XII.1–§XII.5. Compare Milnor-Stasheff Appendix C and Tu Chs. 22–25 [ref: J. Milnor, J. Stasheff, Characteristic Classes (Annals of Math. Studies 76, Princeton University Press 1974) Appendix C — the canonical statement of the Chern-Weil construction in the topology-textbook tradition; L. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes (GTM 275, Springer 2017) Chs. 22–25 — the modern teaching version of KN-II §XII]. Originator-paper citations: S.-S. Chern, "Characteristic classes of Hermitian manifolds," Annals of Math. 47 (1946) 85–121 — the originating paper for the curvature construction of complex characteristic classes; A. Weil, unpublished lecture notes 1949 (circulated as a mimeograph; later collected in Œuvres scientifiques), which extended Chern's curvature recipe to all principal $G$-bundles via invariant polynomials — this is the work that the Codex unit 03.06.06-chern-weil-homomorphism.md is named after.
7. Editorial signature. KN-II is uncompromisingly bundle-and-curvature first in the same spirit as KN-I. Complex manifolds are introduced via the almost-complex / integrability route (not via local holomorphic coordinates as in a complex-analysis text); Kähler manifolds are identified by the Kähler condition $d\omega =0$ after the almost- complex / Chern-connection apparatus is in place; characteristic classes are built from curvature after the full Vol. I + Vol. II §IX curvature calculus is available. This is the opposite order from Wells (which starts from sheaf cohomology and local Dolbeault) and Huybrechts (which starts from local complex geometry); it is the same order as Voisin (which also runs Hodge theory through the curvature-and-connection route) and Tu (which is explicitly a teaching rewrite of KN-II §XII). The principal-bundle-first ordering is what makes KN-II the natural bridge to gauge theory (Yang-Mills measures the curvature norm of a connection on a Hermitian or unitary bundle) and index theory (Atiyah-Singer expresses the index of an elliptic operator as the pairing of characteristic classes against the fundamental class) [ref: Tu Preface; D. Salamon, Spin Geometry and Seiberg-Witten Invariants (preprint, ETH Zürich 2000) §1 — both citing KN-II §XII as the canonical bridge between connection theory and characteristic classes].
8. Peer-source contrast vs Milnor-Stasheff. KN-II §XII and Milnor-Stasheff give two different definitions of characteristic classes that coincide in their cohomological output: KN-II builds $c_i \in H^{2i}{\mathrm{dR}} (M; \mathbb{R})$ from the curvature of any connection on a complex vector bundle (the differential-geometric definition); Milnor-Stasheff builds $c_i\in H^{2i}(M;\mathbb{Z})$ axiomatically and via the classifying-map construction (the homotopy-theoretic / topological definition). The two agree under the de Rham comparison $H^(M; \mathbb{Z}) \otimes \mathbb{R} \cong H^{\mathrm{dR}}(M; \mathbb{R})$ for the Chern classes valued in $\mathbb{R}$ but disagree on torsion (Stiefel-Whitney classes, the integral lift of Chern classes), which is why both viewpoints are needed [ref: Milnor-Stasheff Appendix C; Voisin §11 — the explicit compatibility statement; Bott-Tu §20–§23 — runs both constructions and shows compatibility].

KN-II is not a first textbook on complex manifolds (Wells / Huybrechts serve that role at a more accessible pace) and not a first textbook on characteristic classes (Milnor-Stasheff serves that role with simpler axiomatic foundations). The canonical "before KN-II" sequence is KN-I → Wells → KN-II, or KN-I → Huybrechts → KN-II. The canonical "after KN-II" sequence is KN-II → Voisin (deep Hodge theory) → Donaldson-Kronheimer (four-manifold gauge theory) → Joyce (special holonomy / Calabi-Yau).

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§2 Coverage table (Codex vs KN-II)

Cross-referenced against the current Codex corpus. ✓ = covered, △ = partial / different framing, ✗ = not covered. KN-II material maps primarily to 03-modern-geometry/02-manifolds/, 03-modern-geometry/05-bundles/, 03-modern-geometry/06-characteristic-classes/, the 05-symplectic/almost-complex/ chapter, the 04-algebraic-geometry/09-hodge/ chapter, and adjacent.

KN-II topic	Codex unit(s)	Status	Note
Ch. VIII — submanifolds
Isometric immersion, second fundamental form $\mathrm{II}$	—	✗	Gap. No anchor unit. Cited downstream (e.g. in 05-symplectic/lagrangian/ Lagrangian-submanifold framing) without a definitional unit.
Gauss equation	—	✗	Gap. Foundational; standard prereq for Riemannian comparison geometry.
Codazzi-Mainardi equation	—	✗	Gap.
Ricci equation (normal-bundle curvature)	—	✗	Gap.
Totally geodesic / minimal / totally umbilical submanifolds	—	✗	Gap. Cited informally in symplectic units (Lagrangian = totally null + half-dim) without an anchor.
Ch. IX — complex and almost-complex manifolds
Almost-complex structure $J:TM\to TM$, $J^2=-\mathrm{id}$	05.06.01-almost-complex.md	△	Shipped in the symplectic chapter (not the manifolds chapter), framed as "almost-complex structure on a symplectic manifold". KN-II frames $J$ as a manifold-level structure independent of any symplectic form; the Codex framing is narrower.
Nijenhuis tensor $N_J$	△ (inside 05.06.03)	△	Mentioned in 05.06.03-newlander-nirenberg.md as the integrability obstruction. No dedicated unit; not anchored to a manifold-level $J$.
Type decomposition $TM\otimes \mathbb{C}=T^{1,0}\oplus T^{0,1}$	—	✗	Gap. Foundational; not anchored as its own unit.
Newlander-Nirenberg integrability theorem	05.06.03-newlander-nirenberg.md	✓	Shipped; master anchor Newlander-Nirenberg 1957, Hörmander, Huybrechts, Voisin. Good coverage.
Dolbeault complex $({\Omega }^{p,\ast }(M),\bar{\partial })$	—	✗	Gap. Cited in 05.06.03 Master tier without an anchor unit.
Dolbeault cohomology $H_{\bar{\partial }}^{p,q}(M)$	—	✗	Gap. Load-bearing for Hodge theory and Kodaira-vanishing.
Complex vector bundle	03.05.08-complex-vector-bundle.md	✓	Shipped.
Holomorphic vector bundle (definition, transition functions)	—	✗	Gap. Distinct from smooth complex bundle. Cited in 04.09.02-kodaira-vanishing.md without an anchor.
Hermitian metric on a complex vector bundle	—	✗	Gap.
Chern connection (unique compatible connection on a holomorphic Hermitian bundle)	—	✗	Gap. Distinctive to complex differential geometry; load-bearing for the Chern-class curvature formula.
Ch. IX — Kähler manifolds
Hermitian metric on a complex manifold	—	✗	Gap.
Kähler form $\omega =g(J\cdot ,\cdot )$; Kähler condition $d\omega =0$	—	✗	Gap. Standard reference for the entire Kähler-Hodge-theoretic literature.
Equivalence: $d\omega =0\iff \nabla J=0\iff$ Chern = Levi-Civita	—	✗	Gap. The defining theorem of Kähler geometry.
Kähler identities $[\Lambda ,\partial ]=-i{\bar{\partial }}^{\ast }$ etc.	—	✗	Gap.
Hodge decomposition for compact Kähler manifolds $H^k(\mathbb{C})={⨁}_{p+q=k}{H}^{p,q}$	04.09.01-hodge-decomposition.md	△	A hodge-decomposition unit ships in 04-algebraic-geometry/09-hodge/; it covers the algebraic-geometric framing but may not give the KN-II Kähler-manifold proof. To verify in the production pass.
Kodaira vanishing theorem	04.09.02-kodaira-vanishing.md	✓	Shipped in algebraic geometry chapter; master anchor Kodaira 1954.
$\partial \bar{\partial }$-lemma	—	✗	Gap. Load-bearing for formality of compact Kähler manifolds.
Hard Lefschetz $L^k:H^{n-k}\to H^{n+k}$	—	✗	Gap.
Ch. X — homogeneous spaces
Homogeneous space $G/H$	△	△	Touched in 03-modern-geometry/03-lie/ (Lie-group chapter) and in 05-symplectic/coadjoint/ (coadjoint orbits as $G/G_{\mu }$). No dedicated "homogeneous-space" unit at manifold level.
Reductive decomposition $\mathfrak{g}=\mathfrak{h}\oplus \mathfrak{m}$	—	✗	Gap. Foundational for invariant-connection theory.
Canonical connection on a reductive homogeneous space	—	✗	Gap. Distinctive to KN-II §X.
Nomizu construction (invariant affine connections)	—	✗	Gap.
Symmetric space (Lie-triple condition $[\mathfrak{m},\mathfrak{m}]\subseteq \mathfrak{h}$)	—	✗	Gap. Cited via coadjoint and 03.09.18-berger-holonomy.md (Berger's list includes symmetric spaces) without an anchor.
Cartan classification of irreducible Riemannian symmetric spaces	—	✗	Gap. Referenced from 03.09.18.
Ch. XI — transformations
Holomorphic transformation, infinitesimal holomorphic vector field	—	✗	Gap.
Isometries of Hermitian / Kähler manifolds	—	✗	Gap.
Bochner technique (vanishing theorems via curvature)	—	✗	Gap. Load-bearing for Bonnet-Myers-style results in Kähler setting.
Ch. XII — characteristic classes via Chern-Weil
Invariant polynomial $f\in I^{\ast }(\mathfrak{g})$	03.06.05-invariant-polynomial.md	✓	Shipped.
Chern-Weil homomorphism $I^{\ast }(\mathfrak{g})\to H_{\mathrm{dR}}^{\ast }(M)$	03.06.06-chern-weil-homomorphism.md	✓	Shipped; master anchor "Kobayashi-Nomizu Vol. II Ch. XII". The unit's references block has a source: TODO_REF for the KN-II citation — the production pass should resolve it.
Independence-of-connection theorem	△ (inside 03.06.06)	△	Stated in 03.06.06 Master tier; proof is sketched but not at line-number granularity.
Chern classes $c_i(E)=[\det (I+\frac{i}{2\pi }\Omega )]$ via curvature	03.06.04-pontryagin-chern-classes.md	△	Shipped axiomatically + with ${\mathbb{CP}}^n$ worked example; the explicit curvature formula is in 03.06.06 Master tier but not as a standalone unit.
Pontryagin classes via curvature	03.06.04-pontryagin-chern-classes.md	△	Same as above — axiomatic + reference to Chern-Weil construction; the curvature formula is partial.
Euler class via the Pfaffian of curvature	△	△	Pfaffian computation is referenced in 03.06.06 Master tier but no dedicated unit. The de Rham Euler class is in 03.04.09-thom-global-angular-form.md via the Thom form.
Chern-Simons transgression form	—	✗	Gap. Distinctive to KN-II §XII.4 (and a load-bearing concept for gauge theory and TQFT). Referenced informally in 03.07.05-yang-mills-action.md (Chern-Simons action) without a curvature-side anchor.
Todd class, $\hat{A}$-class, $L$-class via curvature	△	△	$\hat{A}$ appears in 03.09.10 (Atiyah-Singer), $L$ in 03.06.04; the Chern-Weil curvature formulas are partial.
Compatibility of Chern-Weil and Milnor-Stasheff definitions (de Rham comparison)	—	✗	Gap. The unit 03.06.06 references both; the explicit statement and proof of compatibility is not anchored.
Appendices
Integration on complex manifolds (real and complex orientation)	—	✗	Touched implicitly; KN-II Appendix material is short.

Aggregate coverage estimate: ~25–30% of KN-II has corresponding Codex units. Chapter XII (Chern-Weil) is the best-covered chapter at ~60% because 03.06.05 + 03.06.06 + 03.06.04 form the Chern-Weil spine and the Milnor-Stasheff audit has already flagged the remaining gaps. Chapter IX (complex / almost-complex / Kähler) is ~15% covered: Newlander-Nirenberg ships, the almost-complex structure ships (with a symplectic framing), but holomorphic bundles, Chern connection, Hermitian metric, Kähler form, Kähler identities, $\partial \bar{\partial }$-lemma, and hard Lefschetz are all gaps. Chapter VIII (submanifolds) is ~0% covered: no Gauss-Codazzi-Ricci equations, no second fundamental form. Chapter X (homogeneous spaces) is ~5% covered: touched in Lie-group and coadjoint chapters but no canonical-connection unit. Chapter XI (transformations) is ~0% covered: no Bochner technique, no Kähler-isometry framework.

Silent KN-II dependencies in the Codex. The audit reveals a cluster of units that cite KN-II as their master anchor (directly or implicitly) but whose load-bearing prerequisites are uncovered:

1. 03.06.06-chern-weil-homomorphism.md (master = "Kobayashi-Nomizu Vol. II Ch. XII; Milnor-Stasheff Appendix C") — source: TODO_REF for the KN-II citation. Silently depends on a missing Chern connection unit (for the complex-bundle special case) and on the missing transgression / Chern-Simons machinery.
2. 04.09.01-hodge-decomposition.md — covers the algebraic-geometric framing of Hodge decomposition; the KN-II Kähler-manifold proof via Kähler identities is the load-bearing differential-geometric route and has no anchor in the Codex.
3. 04.09.02-kodaira-vanishing.md — cites Kodaira 1954 directly; the prerequisite holomorphic bundle and Hermitian metric / Chern connection layer is missing.
4. 05.06.01-almost-complex.md — frames $J$ as a structure on a symplectic manifold; the KN-II manifold-level framing (independent of any symplectic form) is the wider concept.
5. 05.06.03-newlander-nirenberg.md — cites Voisin and Huybrechts; the prerequisite type decomposition $T^{1,0}\oplus T^{0,1}$ and Dolbeault complex are not anchored.
6. 03.09.18-berger-holonomy.md — Berger's list includes Kähler ($\mathrm{U}(n)$) and Calabi-Yau ($\mathrm{SU}(n)$) holonomy; the prerequisite Kähler-manifold and Calabi-Yau definitions are not anchored.
7. 03.07.05-yang-mills-action.md — Yang-Mills lives on a Hermitian bundle with a $\mathrm{U}(n)$-connection; the prerequisite Hermitian bundle and Chern connection layer overlaps with the gaps above.

Net effect: the existing Codex 06-characteristic-classes/ chapter cites KN-II as its canonical reference and covers ~60% of the relevant material; the 02-manifolds/ chapter covers ~10% of the complex / Kähler prerequisite layer; the 04-algebraic-geometry/09-hodge/ chapter covers the Hodge / Kodaira downstream at ~70% via algebraic-geometric framing but with the differential-geometric KN-II proof route absent. The KN-II audit is therefore load-bearing across four chapters: 02-manifolds/, 05-bundles/, 06-characteristic-classes/, and 04.09-hodge/.

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

Priority 0 — prerequisite chain shared with KN-I and Milnor-Stasheff audits:

These are listed here because KN-II assumes them; they are already on the KN-I punch-list (FT 3.18) and / or the Milnor-Stasheff punch-list (FT 3.08). Ship once, cite from both.

• 03.05.14 Torsion tensor and the two Cartan structural equations (KN-I punch-list, priority 1) — load-bearing for KN-II §IX (Chern connection is the unique connection with $T^{0,2}=0$ on a holomorphic Hermitian bundle).
• 03.03.X1 Levi-Civita connection, geodesics, exp, Hopf-Rinow (KN-I punch-list, priority 1) — load-bearing for KN-II §VIII (submanifold geometry) and §IX (Kähler ⇔ Chern = Levi-Civita).
• 03.05.13 Associated bundle and induced connection (KN-I punch-list, priority 1) — load-bearing for KN-II §XII (curvature of an associated vector bundle is what enters Chern-Weil).

Priority 1 — Ch. IX core (complex / almost-complex / Hermitian / Kähler):

1. 03.02.09 Almost-complex structure (manifold-level). $J:TM\to TM$ with $J^2=-\mathrm{id}$; type decomposition $TM \otimes \mathbb{C} = T^{1,0}M \oplus T^{0,1}M$;theNijenhuistensor$N_J(X, Y)$. Sits in 02-manifolds/ (per the KN-I audit, this chapter is wide open). This is a manifold-level lift of the existing 05.06.01-almost-complex.md (which lives in the symplectic chapter and uses the symplectic framing). Cross-reference both. KN-II §IX.1–§IX.2 anchor; Wells §I.1 + Huybrechts §2.6 secondary. ~1500 words. Three-tier.
2. 03.02.10 Complex manifold and the Dolbeault complex. Holomorphic atlas; ${\Omega }^{p,q}(M)$ as smooth $(p,q)$-forms; the operators $\partial, \bar\partial$with$\bar\partial^2 = 0$; the Dolbeault complex $({\Omega }^{p,\ast },\bar{\partial })$ and Dolbeault cohomology $H_{\bar{\partial }}^{p,q}(M)$. KN-II §IX.2–§IX.3 anchor; Wells Ch. II + Huybrechts §2.6 secondary. ~1800 words. Three-tier.
3. 03.05.19 Holomorphic vector bundle. Definition (transition functions holomorphic), Dolbeault ${\bar{\partial }}_E$ on ${\Omega }^{p,\ast }(E)$, the Koszul-Malgrange theorem that holomorphic structures on a smooth complex bundle ↔ flat $\bar{\partial }$-operators. KN-II §IX.4 anchor; Wells §III.1 secondary; Huybrechts §2.6 secondary. ~1500 words. Three-tier. Closes the silent gap behind 04.09.02-kodaira-vanishing.md.
4. 03.05.20 Hermitian metric on a complex bundle; Chern connection. Hermitian metric $h$ on a complex bundle $E$; the unique connection ${\nabla }^h$ compatible with both the holomorphic structure ${\bar{\partial }}_E$ and the Hermitian metric (the Chern connection). Formula in local holomorphic frame: ${\nabla }^h=\partial +h^{-1}\partial h$. KN-II §IX.5 anchor; Wells §III.2; Huybrechts §4.2 secondary. ~1500 words. Three-tier. Closes the silent gap behind 03.06.06-chern-weil-homomorphism.md for the complex-bundle special case.
5. 03.02.11 Hermitian manifold and the Kähler form. Hermitian metric $g$ on $(M,J)$ with $g(JX,JY)=g(X,Y)$; the Kähler form $\omega \in \Omega^{1,1}(M)$definedby$\omega(X, Y) = g(JX, Y)$; Kähler condition $d\omega =0$. The equivalence theorem: $d\omega = 0 \Leftrightarrow \nabla J = 0 \Leftrightarrow$Chernconnectionon$TM$ equals the Levi-Civita connection. KN-II §IX.6–§IX.7 anchor; Huybrechts §3.1 secondary; Voisin §3.1 secondary. ~2000 words. Three-tier. Master tier of any Hodge-theoretic unit downstream depends on this.
6. 03.02.12 Kähler identities and the Hodge decomposition (Kähler version). The Kähler identities $[\Lambda ,\partial ]=-i{\bar{\partial }}^{\ast }$, $[\Lambda ,\bar{\partial }]=i{\partial }^{\ast }$; the consequence that ${\Delta }_d=2{\Delta }_{\bar{\partial }}=2{\Delta }_{\partial }$ on a compact Kähler manifold; the Hodge decomposition $H^k(M;\mathbb{C})={⨁}_{p+q=k}{H}^{p,q}(M)$ with $H^{p,q} = \overline{H^{q,p}}$. KN-II §IX.7 anchor; Voisin §6 secondary; Huybrechts §3.2 secondary. ~2000 words. Master-tier dominant; Intermediate gives the statement of Hodge decomposition only. Sharpens 04.09.01-hodge-decomposition.md with the differential-geometric proof route.

Priority 2 — Ch. XII Chern-Weil deepenings + Ch. VIII submanifolds:

7. 03.06.07 Chern-Simons / transgression form. For a connection $\omega$ on a principal $G$-bundle with curvature $\Omega$, the transgression form $TP(\omega )\in {\Omega }^{2k-1}(P)$ with $dTP(\omega )=P({\Omega }^k)$ for an invariant polynomial $P$ of degree $k$. The Chern-Simons 3-form $CS(\omega) = \mathrm{tr}(\omega \wedge d\omega + \tfrac{2}{3} \omega \wedge \omega \wedge \omega)$ as the canonical transgression of the second Chern class. KN-II §XII.4 anchor; Chern-Simons 1974 originator citation. ~1500 words. Master-tier dominant. Bridges 03.06.06-chern-weil-homomorphism.md to 03.07.05-yang-mills-action.md (Chern-Simons action) and to topological field theory.
8. 03.06.18 Compatibility of Chern-Weil and Milnor-Stasheff Chern classes. The de Rham comparison: the curvature class $[\frac{i}{2\pi }\mathrm{tr}(\Omega )]\in H_{\mathrm{dR}}^2(M;\mathbb{R})$ equals the image of the topological $c_1(E)\in H^2(M;\mathbb{Z})$ under $H^{\ast }(M;\mathbb{Z})\to H^{\ast }(M;\mathbb{R})\cong H_{\mathrm{dR}}^{\ast }(M;\mathbb{R})$. Generalisation to all Chern classes. KN-II §XII.5 anchor; Milnor-Stasheff Appendix C secondary. ~1500 words. Master-tier dominant. Shared with the Milnor-Stasheff punch-list (FT 3.08, priority-3); ship once and cite from both audits.
9. 03.02.13 Isometric immersion and the second fundamental form. Isometric immersion $f:M\hookrightarrow \bar{M}$ of Riemannian manifolds; the second fundamental form $\mathrm{II}:TM\times TM\to NM$; the mean curvature vector $H=\mathrm{tr}\mathrm{II}$. KN-II §VIII.1 anchor; do Carmo Ch. 6 secondary. ~1500 words. Three-tier.
10. 03.02.14 Gauss, Codazzi, and Ricci equations. The three structure equations relating the curvature of $M$, the curvature of $\bar{M}$, and the second fundamental form. Worked example: $S^n \subseteq \mathbb{R}^{n+1}$ with the standard immersion. KN-II §VIII.2–§VIII.4 anchor; do Carmo Ch. 6 secondary. ~2000 words. Three-tier.

Priority 3 — Ch. X homogeneous spaces + Ch. XI transformations:

11. 03.03.X2 Homogeneous space $G/H$ and reductive decomposition. Quotient manifold structure on $G/H$ for $G$ a Lie group and $H$ a closed subgroup; reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$with$\mathrm{Ad}(H)\mathfrak{m} \subseteq \mathfrak{m}$. KN-II §X.1–§X.2 anchor; Helgason Ch. IV secondary. ~1500 words. Three-tier. Sits in the 03-lie/ chapter; lateral connection to 05-symplectic/coadjoint/ (coadjoint orbits as homogeneous spaces).
12. 03.03.X3 Canonical connection on a reductive homogeneous space; symmetric space. The Nomizu construction of $G$-invariant affine connections; the canonical connection (corresponding to $\mathrm{Ad}(H)$- invariant bilinear form on $\mathfrak{m}$); specialisation to symmetric spaces ($[\mathfrak{m},\mathfrak{m}]\subseteq \mathfrak{h}$). Pointer to the Cartan classification of irreducible Riemannian symmetric spaces (statement only). KN-II §X.2–§X.6 anchor; Helgason Chs. IV, VIII secondary. ~1800 words. Master-tier dominant.
13. 03.02.15 Bochner technique and curvature vanishing theorems. Weitzenböck formula on a Riemannian manifold; vanishing of harmonic forms and Killing fields under curvature-positivity assumptions; the Kähler-manifold specialisation (vanishing of $H^{p,0}$ on a manifold with positive Ricci form). KN-II §XI.4–§XI.5 anchor; Lawson-Michelsohn §II.8 secondary. ~1500 words. Master-tier dominant.

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

14. Pointer in 03.06.06-chern-weil-homomorphism.md to the new 03.05.20 (Chern connection) and 03.06.07 (Chern-Simons) units, and resolve the source: TODO_REF to KN-II §XII with concrete section locators. Single-paragraph weaving edit; resolves the audit-flagged TODO_REF.
15. Pointer in 04.09.01-hodge-decomposition.md to the new 03.02.12 (Kähler identities + Kähler Hodge decomposition) as the differential-geometric route to the same theorem. Single-paragraph weaving edit.
16. Pointer in 04.09.02-kodaira-vanishing.md to the new 03.05.19 (holomorphic bundle) and 03.05.20 (Hermitian metric / Chern connection) as the prerequisite layer.
17. Pointer in 05.06.01-almost-complex.md to the new 03.02.09 (manifold-level almost-complex structure) as the upstream generic framing; the symplectic specialisation in 05.06.01 becomes a canonical example of the generic concept.
18. Pointer in 05.06.03-newlander-nirenberg.md to 03.02.09 and 03.02.10 (Dolbeault complex) as the prerequisite layer; resolve the TODO_REF blocks accordingly.
19. Pointer in 03.09.18-berger-holonomy.md to 03.02.11 (Kähler manifold) for the $\mathrm{U}(n)$ holonomy case and a forward pointer to the (deferred) Calabi-Yau unit for $\mathrm{SU}(n)$.
20. Pointer in 03.07.05-yang-mills-action.md to 03.06.07 (Chern-Simons) for the Chern-Simons action specialisation.

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§4 Implementation sketch (P3 → P4)

For a full KN-II coverage pass, priority-0 + priority-1 are the minimum set and close the complex-and-Chern-Weil silent-dependency gaps across four chapters. Realistic production estimate (mirroring earlier Brown / Bott-Tu / Milnor / KN-I batches):

• ~3.5 hours per unit. KN-II units skew slightly above the corpus average (and the KN-I average) because the Master tier requires careful juggling of (i) the principal-bundle calculus from Vol. I, (ii) the complex-bundle / holomorphic / Hermitian layer, and (iii) the curvature-formula bookkeeping (Pfaffians, determinants of $\Omega$).
• Priority 0: 3 units shared with the KN-I + Milnor-Stasheff punch-lists (do not double-count); flagged here for prerequisite tracking only.
• Priority 1: 6 units × ~3.5 hours = ~21 hours. Heart of the audit; closes the complex / Kähler gap in one batch and supplies prerequisites for the Hodge chapter and the Chern-Weil unit.
• Priority 2: 4 units × ~3.5 hours = ~14 hours. Closes the Chern-Simons / transgression gap, the de Rham–topological compatibility gap, and the submanifold-equations gap (Ch. VIII).
• Priority 3: 3 units × ~3 hours = ~9 hours. Closes Ch. X (homogeneous spaces) and Ch. XI (Bochner technique).
• Priority 4: 7 weaving edits × ~30 min = ~3.5 hours.
• Total for full coverage: ~47–48 hours, roughly a focused 7–10 day window. Priority-0 (carried by other audits) + priority-1 alone is ~21 hours and closes the load-bearing complex / Kähler gaps; with priority-2 (~14 hours more) the Chern-Weil chapter reaches FT-equivalence.

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

• S.-S. Chern, "Characteristic classes of Hermitian manifolds," Annals of Math. 47 (1946) 85–121 — the originating paper for the curvature construction of complex characteristic classes. Cite in 03.06.06 (audit-flagged TODO_REF resolution) and in 03.05.20 (Chern connection — historically derived from this paper).
• André Weil, unpublished 1949 lecture notes (mimeograph; later in Œuvres scientifiques Vol. III) — generalised Chern's curvature recipe to all principal $G$-bundles via invariant polynomials. Cite in 03.06.06 as the second half of the "Chern-Weil" name.
• A. Newlander, L. Nirenberg, "Complex analytic coordinates in almost complex manifolds," Annals of Math. 65 (1957) 391–404 — the integrability theorem. Already cited in 05.06.03 (TODO_REF block); resolve concretely during the production pass and cross-cite in 03.02.09 (manifold-level almost-complex structure).
• K. Kodaira, "On Kähler varieties of restricted type," Annals of Math. 60 (1954) 28–48 — the foundational paper for the compact-Kähler- manifold programme (Kodaira embedding, projectivity criterion). Cite in 03.02.11 (Kähler manifolds) and in 04.09.02 (Kodaira vanishing — already cited there).
• S.-S. Chern, J. Simons, "Characteristic forms and geometric invariants," Annals of Math. 99 (1974) 48–69 — the originating paper for the Chern-Simons transgression form. Cite in 03.06.07.
• W. V. D. Hodge, The Theory and Applications of Harmonic Integrals (Cambridge University Press 1941, 2nd ed. 1952) — originating the Hodge decomposition. Cite in 03.02.12.
• K. Nomizu, "Invariant affine connections on homogeneous spaces," Amer. J. Math. 76 (1954) 33–65 — the Nomizu construction. Cite in 03.03.X3.
• S. Bochner, "Vector fields and Ricci curvature," Bull. AMS 52 (1946) 776–797 — the Bochner technique. Cite in 03.02.15.
• Kobayashi, Nomizu (1969) — the canonical consolidation. Cite throughout as the in-Codex master anchor.

Notation crosswalk. KN-II writes:

• $J$ for the almost-complex structure; $N_J$ for the Nijenhuis tensor.
• $T^{1,0}$ and $T^{0,1}$ for the type-$(1,0)$ and type-$(0,1)$ subbundles of $T_{\mathbb{C}}M$.
• ${\Omega }^{p,q}(M)$ for the smooth $(p,q)$-forms.
• $\partial ,\bar{\partial }$ for the Dolbeault operators; $H_{\bar{\partial }}^{p,q}(M)$ for Dolbeault cohomology.
• $h$ for a Hermitian metric on a complex vector bundle; ${\nabla }^h$ for the Chern connection.
• $\omega$ for the Kähler form (reused from KN-I where $\omega$ is the connection 1-form — notation clash to flag in the Codex; resolve by writing ${\omega }_K$ for the Kähler form when in the same paragraph as the connection 1-form, or by using ${\omega }^{\mathrm{conn}}$ for the latter, as Tu does).
• $I^{\ast }(\mathfrak{g})$ for the algebra of $\mathrm{Ad}$-invariant polynomials on $\mathfrak{g}$.
• $w:I^{\ast }(\mathfrak{g})\to H_{\mathrm{dR}}^{\ast }(M)$ for the Chern-Weil homomorphism.
• $c_i,p_i,e,\mathrm{Td},\hat{A},L$ for the characteristic classes (same as Milnor-Stasheff and Codex existing convention).

Wells, Huybrechts, and Voisin use the same complex-geometry notation. Tu uses identical notation to KN-II in the bundle / curvature register; Tu adopts the $\omega \leftrightarrow A$, $\Omega \leftrightarrow F$ aliases to bridge to the gauge-theory notation in 03.07.05-yang-mills-action.md (already recorded in the KN-I plan §4). The Codex notation decision (per docs/specs/UNIT_SPEC.md §11) should: adopt KN-II's complex-geometry notation verbatim; resolve the $\omega$ notation clash by reserving $\omega$ for the Kähler form inside the complex / Kähler units (03.02.11, 03.02.12) and writing ${\omega }^{\mathrm{conn}}$ in those units when the principal-bundle connection 1-form appears explicitly. Record in a §Notation paragraph of each affected unit.

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§5 What this plan does NOT cover

• A line-number-level inventory of every named theorem in KN-II (full P1 audit). Deferred until a local PDF is on disk in reference/fasttrack-texts/03-modern-geometry/Kobayashi-Nomizu-FoundationsVol2.pdf. This stub works from canonical TOC knowledge + peer-source crosswalks
  • Codex internal evidence and is therefore REDUCED.
• The deep Hodge-theoretic programme (Lefschetz decomposition, primitive cohomology, polarised Hodge structures, mixed Hodge structures, period domains, period maps, the Calabi-Yau Theorem, the Hodge conjecture). Fast Track 3.27 — Voisin, Hodge Theory and Complex Algebraic Geometry I + II — is the canonical Codex audit target for that material. KN-II §IX only goes as far as Hodge decomposition on compact Kähler manifolds; everything downstream is Voisin territory and is deferred to its own audit.
• Kähler-Einstein metrics and the Calabi-Yau theorem. KN-II does not cover Yau's solution of the Calabi conjecture. Deferred to a future Joyce / Tian audit pass (no current Fast Track entry).
• Sasaki-Einstein and special holonomy beyond Kähler. Cited via 03.09.18-berger-holonomy.md; deferred to a future Joyce audit.
• Algebraic-geometry-side cohomology of Kähler manifolds (Lefschetz hyperplane theorem, Bertini, Kodaira-Akizuki-Nakano vanishing, Kodaira embedding, projective embedding of Kähler varieties of restricted type). Sits in the 04-algebraic-geometry/ chapter and is partly shipped (04.09.02-kodaira-vanishing.md); not on this audit's scope.
• Index theory for the Dolbeault operator (Riemann-Roch, Hirzebruch-Riemann-Roch, Atiyah-Singer for $\bar{\partial }$). Touched in 03.09.10 (Atiyah-Singer) and 04-algebraic-geometry/04-riemann-roch/; not on this audit's scope.
• Yang-Mills / instanton theory specifics (ASD equations, Donaldson invariants, ADHM construction). Fast Track 3.20 (Donaldson- Kronheimer / Atiyah-Bott) is the canonical audit target. Deferred.
• The $G_2$, $\mathrm{Spin}(7)$, and Calabi-Yau holonomy specialisations. Joyce-territory; deferred.

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§6 Acceptance criteria for FT equivalence (KN-II)

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

• ≥95% of KN-II's named theorems in Chs. VIII–XII map to Codex units. Current ~25–30%; after priority-1 rises to ~65%; after priority-1+2 to ~85%; after priority-1+2+3 to ~93%; full ≥95% requires priority-4 weaving (no new units, only cross-references).
• ≥90% of KN-II's worked computations in Chs. IX + XII (the load-bearing chapters) have either a direct unit or a worked example in a cross-referenced unit. Canonical KN-II computations to cover: ${\mathbb{CP}}^n$ as a Kähler manifold; the Hopf bundle as a holomorphic line bundle and its first Chern class; the Fubini-Study metric; the Pfaffian computation of $e(T{S}^{2n})$; the second Chern class of an $\mathrm{SU}(2)$-instanton over $S^4$.
• The source: TODO_REF for KN-II §XII in 03.06.06-chern-weil-homomorphism.md is resolved to a concrete chapter/section locator.
• The TODO_REF blocks in 05.06.01-almost-complex.md and 05.06.03-newlander-nirenberg.md for the complex-geometry secondary references (Cannas da Silva, Huybrechts, Voisin) are resolved.
• The Milnor-Stasheff-shared item (03.06.18 Chern-Weil / Milnor-Stasheff compatibility) ships once and is cited from both audits without duplication.
• The KN-I-shared items (priority-0 above) ship once and are cited from both audits.
• 03.09.18-berger-holonomy.md and 04.09.01-hodge-decomposition.md and 04.09.02-kodaira-vanishing.md and 05.06.01-almost-complex.md and 05.06.03-newlander-nirenberg.md carry weaving paragraphs pointing to the new complex / Kähler / Chern-Weil units.
• Notation decisions are recorded in 03.02.09, 03.02.10, 03.02.11, 03.02.12, 03.05.19, 03.05.20, 03.06.07, 03.06.18 (especially the $\omega$ Kähler-form vs $\omega$ connection-1-form clash resolution; see §4).
• Originator-prose citations of Chern 1946, Weil 1949, Newlander-Nirenberg 1957, Kodaira 1954, Chern-Simons 1974, Hodge 1941, Nomizu 1954, and Bochner 1946 are present in the relevant units.

The 6 priority-1 units close the complex / Kähler / Chern-connection silent-dependency gaps and lift KN-II coverage from ~25% to ~65%. Priority-2 (4 units) closes the Chern-Weil deepenings (transgression, Milnor-Stasheff compatibility) and the submanifold-equations gap. Priority-3 (3 units) closes the homogeneous-space / Bochner gaps. Priority-4 is weaving only and does not add new units.

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§7 Sourcing

• Not free. KN-II is under active Wiley copyright (1969, reprint 1996). Same legal status as KN-I — no free legal PDF is hosted by the authors' institutions or by Wiley. Print copies (Wiley Classics Library reprint, ISBN 0-471-15732-5) are readily available second-hand. Library mirror or Anna's Archive likely needed for a local copy; the Fast Track entry currently holds only the cover-image stub already shared with KN-I (reference/fast-track/images/Kobayashi-Nomizu-1-683x1024__72960fe9e3.jpg) — note that this stub may show only the Vol. I cover; the Vol. II cover image is not in the archive at the time of this audit.
• Local copy. Not present in reference/textbooks-extra/ or reference/fasttrack-texts/03-modern-geometry/ at the time of this audit. A scanned copy should be added to reference/fasttrack-texts/03-modern-geometry/Kobayashi-Nomizu-FoundationsVol2.pdf before the production pass on the priority-1 punch-list units, so that line-number citations can be resolved in the TODO_REF blocks of 03.06.06 and in the new units that will cite KN-II Chs. IX + XII at master tier.
• Companion peer texts (cited in §1):
  • R. O. Wells, Differential Analysis on Complex Manifolds, GTM 65, Springer 3rd ed. 2008 (with new appendix by O. García-Prada). The standard graduate text on the analytic side of complex geometry; Chs. I–III parallel KN-II §IX with a sheaf-theoretic rather than principal-bundle framing. The preferred secondary anchor for 03.02.10, 03.05.19, 03.05.20.
  • D. Huybrechts, Complex Geometry: An Introduction, Universitext, Springer 2005. The most accessible modern graduate text; Ch. 2 covers almost-complex / complex / Hermitian / Chern connection, Ch. 3 covers Kähler manifolds, Ch. 4 covers Hodge theory and Lefschetz decomposition, Ch. 5 covers Kähler-Einstein and Calabi-Yau. The preferred secondary anchor for 03.02.09–03.02.12 and 03.05.19–03.05.20.
  • C. Voisin, Hodge Theory and Complex Algebraic Geometry I + II, Cambridge Studies in Advanced Mathematics 76 + 77, CUP 2002–2003. The canonical modern treatment of Hodge theory on compact Kähler manifolds. Fast Track 3.27. Voisin §3 + §6–§8 directly parallels KN-II §IX.7 + Hodge decomposition; deeper Hodge-theoretic material is deferred to the FT 3.27 audit (see §5).
  • L. Tu, Differential Geometry: Connections, Curvature, and Characteristic Classes, GTM 275, Springer 2017. The most accessible modern rewriting of KN-I + KN-II §XII; Chs. 22–25 are a teaching rewrite of the Chern-Weil construction. The preferred secondary anchor for 03.06.06, 03.06.07, 03.06.18 (alongside KN-II).
  • J. Milnor, J. Stasheff, Characteristic Classes, Annals of Math. Studies 76, Princeton University Press 1974. Fast Track 3.08. The canonical topological / classifying-space construction of characteristic classes — the contrast to KN-II §XII. Appendix C of Milnor-Stasheff is the explicit bridge to Chern-Weil. Audit shipped this Cycle 4 (plans/fasttrack/milnor-stasheff-characteristic-classes.md); coordinate on the 03.06.18 compatibility unit.
  • S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, Graduate Studies in Mathematics 34, AMS 2001. The standard reference for homogeneous and symmetric spaces; Chs. IV–VIII parallel KN-II §X. Secondary anchor for 03.03.X2, 03.03.X3.
  • M. P. do Carmo, Riemannian Geometry, Birkhäuser 1992. Already cited in the KN-I plan; Ch. 6 covers submanifold geometry (Gauss-Codazzi-Ricci) at a gentler pace than KN-II §VIII. Secondary anchor for 03.02.13, 03.02.14.
  • H. B. Lawson, M.-L. Michelsohn, Spin Geometry, Princeton Mathematical Series 38, Princeton University Press 1989. Fast Track 3.10. §II.8 covers the Bochner technique in the spin / Dirac-operator setting. Secondary anchor for 03.02.15.
• Originator-paper archive locations:
  • Chern 1946 Annals — JSTOR.
  • Weil 1949 lectures — collected in Œuvres scientifiques Vol. III (Springer 1979); print only.
  • Newlander-Nirenberg 1957 Annals — JSTOR.
  • Kodaira 1954 Annals — JSTOR.
  • Chern-Simons 1974 Annals — JSTOR.
  • Hodge 1941 Harmonic Integrals — Cambridge University Press; print only.
  • Nomizu 1954 Amer. J. Math. — JHU Press / JSTOR.
  • Bochner 1946 Bull. AMS — AMS journals.

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§8 Coordination with sibling audits (Cycle 4)

The KN-II audit overlaps three sibling audits shipped or in flight this Cycle 4:

1. KN-I (FT 3.18) — plans/fasttrack/kobayashi-nomizu-foundations-vol1.md. Strict prerequisite chain. The KN-II priority-0 units (torsion / Cartan structural equations, Levi-Civita / Hopf-Rinow, associated bundle / induced connection) are already on the KN-I priority-1 punch-list. Production order: KN-I priority-1 must precede KN-II priority-1. The KN-II Chern-connection unit (03.05.20) is the complex-bundle specialisation of the KN-I 03.05.13 (associated bundle and induced connection); cross-reference both.
2. Milnor-Stasheff (FT 3.08) — plans/fasttrack/milnor-stasheff-characteristic-classes.md. The 03.06.18 (Chern-Weil ↔ topological compatibility) unit is shared. Milnor-Stasheff approaches characteristic classes from the homotopy / classifying-space side; KN-II approaches them from the curvature side. The compatibility unit is the bridge and should ship as a co-authored unit citing both anchors at master tier.
3. Voisin (FT 3.27) — not yet audited. Voisin is the deep Hodge-theoretic sequel to KN-II §IX. The KN-II priority-1 unit 03.02.12 (Kähler identities + Hodge decomposition) is the upstream of Voisin's programme; Voisin's audit should treat 03.02.12 as a prerequisite anchor and pick up at Lefschetz decomposition / primitive cohomology / polarised Hodge structures.

The three audits together close the complex-and-characteristic-classes spine of the modern-geometry chapter (FT 3.08 + 3.18 + 3.19 + 3.27). The Chern-Weil / Chern-connection / Kähler-Hodge subchain is the load-bearing centre of that spine, and the priority-1 + priority-2 units of this KN-II audit are the most-cited new units across all four sibling-audit punch-lists.