Line bundles and their curvature: what 'positive' means
Guides 3 and 4 gave us the Kähler world and its Hodge theory; now we ask what it is good FOR. The answer is that a single number-valued gadget — a holomorphic line bundle L on a compact Kähler manifold M — can be 'positive', and positivity is enough to drag M bodily into projective space. First fix the object. A holomorphic line bundle is a family of one-dimensional complex vector spaces L_p, one over each point p in M, glued by holomorphic transition functions; its sections are the natural generalization of holomorphic functions, but functions that may twist as you move around M. The trivial bundle has global sections that are just holomorphic functions; on a compact M those are only constants, so the interesting bundles are the twisted ones.
To talk about curvature we need a connection. From the rung on bundles you know a Hermitian metric h on L together with its compatible Chern connection produces a curvature 2-form, and for a line bundle that curvature is a closed real (1,1)-form Theta. Dividing by 2 pi gives a de Rham class that does not depend on the metric h chosen: it is the first Chern class c_1(L), an element of H^2(M; R) (in fact H^2(M; Z)). The class c_1(L) is the topological shadow of L; the form Theta is one analytic representative of it, and changing the metric h only changes Theta by an exact i partial dbar(something). This freedom to reshape Theta within its class is exactly what we will exploit.
Now the definition that organizes everything. The line bundle L is positive if it admits a Hermitian metric whose curvature form Theta is a Kähler form — that is, the (1,1)-form (i/2 pi) Theta is positive-definite at every point, so it serves as the omega of a Kähler metric on M. Equivalently, the class c_1(L) contains a Kähler form; such a class is called a positive (or ample) class. The leading example: on CP^n the hyperplane bundle O(1) is positive, its curvature being exactly the Fubini-Study form omega_FS you met in Guide 3. Positivity is a curvature-positivity condition, the line-bundle analogue of positive Ricci curvature, and it is the hypothesis on which the whole embedding theorem will hinge.
Kodaira vanishing: where positivity does its work
Before we can embed anything we need to know that a positive bundle has enough sections to see by. The supply of sections is governed by cohomology, and the engine that guarantees the supply is the Kodaira vanishing theorem: if L is a positive line bundle on a compact Kähler manifold M of complex dimension n, then H^q(M, K_M tensor L) = 0 for every q greater than 0, where K_M is the canonical bundle (the top exterior power of T*M, the bundle of holomorphic n-forms). The slogan 'higher cohomology vanishes' hides where the work happens — and the proof is a Bochner-type argument straight out of the Hodge machinery of Guide 4.
Here is the shape of why it is true, kept honest as a sketch. By the Hodge theorem a class in H^q(M, K_M tensor L) is represented by a unique harmonic form alpha, killed by the bundle Laplacian Delta_dbar built from the Chern connection of L. The Kähler identities of Guide 3 rewrite this Laplacian via a Bochner-Kodaira-Nakano formula as Delta_dbar = Delta_partial + (a curvature term). The curvature term is built from Theta, the curvature of L. When L is positive, Theta is a Kähler form, and a short linear-algebra computation shows the curvature term is a STRICTLY positive operator on forms of degree q greater than 0. A harmonic alpha satisfies 0 = (Delta_dbar alpha, alpha) = (positive part) + (nonnegative part), forcing alpha = 0. Positivity of curvature becomes positivity of an operator becomes vanishing of cohomology.
Kodaira embedding: from a positive bundle to projective space
Now the headline of the rung. The Kodaira embedding theorem says: a compact complex manifold M admits a positive holomorphic line bundle if and only if M can be holomorphically embedded into some projective space CP^N. Equivalently, M is a projective variety — the zero locus of finitely many homogeneous polynomials — precisely when it carries a positive line bundle. This is a stunning bridge: positivity is an analytic, differential-geometric condition about curvature, while being projective is an algebraic condition about polynomial equations. Kodaira's theorem says they are the same condition wearing two costumes. It is the gateway through which complex differential geometry walks into algebraic geometry.
How does one actually build the embedding? You do not use L itself but a high power L^k for k large, and you use its global holomorphic sections to write down a map. Suppose H^0(M, L^k) has a basis s_0, s_1, ..., s_N of global sections. At a point p where not all sections vanish, send p to the point [s_0(p) : s_1(p) : ... : s_N(p)] of CP^N. The values s_j(p) live in the one-dimensional fiber L^k_p, so individually they are not numbers, but their RATIOS are well defined — and a point of projective space is exactly a tuple up to common scaling. This is the map Phi_{L^k}: M -> CP^N. Everything now rides on whether Phi is defined everywhere, injective, and an immersion.
- Base-point freeness: for k large enough, at every point p some section in H^0(M, L^k) is nonzero, so the map Phi has no point where all coordinates vanish — it is defined on all of M. Kodaira vanishing is what guarantees enough sections exist.
- Injectivity (separating points): for k large, given two distinct points p and q there is a section vanishing at p but not at q, so Phi(p) and Phi(q) differ — Phi is one-to-one.
- Immersion (separating tangents): for k large, sections separate first-order data too, so the differential of Phi is injective at every point — no tangent directions get crushed. A compact injective immersion is automatically an embedding.
- Each of these is proved by applying Kodaira vanishing to L^k twisted by the ideal sheaf of the relevant points; the vanishing makes a restriction map surjective, which is exactly 'a section with the prescribed values exists'. Positivity feeds all three.
Calabi-Yau manifolds: when the canonical bundle is flat
Kodaira embedding cares about a positive line bundle. Calabi-Yau geometry cares about the opposite extreme of one special bundle: the canonical bundle K_M, the bundle of holomorphic n-forms, being as flat and trivial as possible. A compact Kähler manifold M is Calabi-Yau when its canonical bundle is trivial, K_M = O_M — equivalently, when c_1(M) = 0 and M carries a nowhere-vanishing holomorphic n-form Omega. Triviality of K_M says the manifold has, in a precise sense, no preferred volume twisting; the holomorphic n-form Omega is a global complex volume element, unique up to a scalar. The simplest Calabi-Yau of each dimension makes this concrete: in dimension one it is the elliptic curve C / Lambda, whose holomorphic 1-form dz never vanishes.
The reason these manifolds matter is the Calabi conjecture, proved by Yau, which is a statement about Ricci curvature. On a Kähler manifold the Ricci curvature is itself represented by a (1,1)-form whose class is exactly 2 pi c_1(M); so c_1(M) = 0 means the Ricci class vanishes. Yau's theorem says far more than 'the class vanishes': given c_1(M) = 0, there exists a genuine Kähler metric whose Ricci curvature is identically zero — a Ricci-flat Kähler metric — and it is unique in each Kähler class. So a compact Kähler manifold with c_1 = 0 admits, for free, a metric that is simultaneously Kähler and a vacuum solution of the Einstein equation. This is the only general existence theorem for Ricci-flat metrics in any dimension, and it is why physicists adopted Calabi-Yau threefolds as the hidden dimensions of string theory.
Be careful with the word 'proof' here. Yau's resolution of the Calabi conjecture reduces existence of the Ricci-flat metric to solving a complex Monge-Ampère equation — a fully nonlinear second-order PDE for a single function phi, schematically det(g_jk + partial_j partial_k phi) = e^F det(g_jk). Yau established the a priori estimates that make this solvable; the argument is a long and delicate chapter of geometric analysis, not a paragraph. The picture below records what the equation looks like, but the genuine proof is a graduate course in nonlinear PDE, not a guide.
Calabi-Yau, three equivalent faces of one condition on a compact Kahler M:
topology : c_1(M) = 0 in H^2(M; R)
holomorphy : K_M trivial, i.e. a nowhere-zero holomorphic n-form Omega exists
metric : there is a UNIQUE Ricci-flat Kahler metric in each Kahler class (Yau)
The Ricci-flat metric solves a complex Monge-Ampere equation for one function phi:
det( g_jk + d_j d_k phi ) = e^F * det( g_jk ), omega_phi = omega + i ddbar phi > 0
Lowest-dimensional Calabi-Yau manifolds:
dim 1 : elliptic curve C / Lambda (1-form dz never vanishes)
dim 2 : K3 surface (e.g. a quartic in CP^3)
dim 3 : the quintic threefold in CP^4 (zero set of a degree-5 polynomial)Examples, open problems, and where the rung leads
Let the examples carry the understanding, exactly as Volume I urged. The quintic threefold — the zero set of a generic degree-5 homogeneous polynomial in CP^4 — is the textbook Calabi-Yau threefold, and you can see both theorems at work on it. It is projective by construction, so by Kodaira embedding it carries a positive line bundle (the restriction of O(1)). A short adjunction computation shows its canonical bundle is trivial, so c_1 = 0, and Yau's theorem then hands it a Ricci-flat Kähler metric — a metric no one can write down in closed form, yet whose existence is certain. The K3 surface (dimension two) is the analogue one dimension lower; the elliptic curve is the analogue one dimension lower still. Same three faces, increasing richness.
Be honest about what remains open. The full classification of Calabi-Yau threefolds is unknown — we do not even know whether there are finitely many topological types, and the count of known ones runs into the hundreds of thousands. The deepest organizing idea is mirror symmetry, which we can only state: it conjectures that Calabi-Yau threefolds come in mirror pairs (M, M') whose Hodge numbers are swapped, h^{p,q}(M) = h^{n-p,q}(M'), and whose complex and symplectic geometries are interchanged. This is a survey statement, not a theorem proved here; large parts of it are still conjectural, and where it has been made rigorous the proofs are formidable. Treat the Hodge symmetry of Guide 4 as the warm-up and mirror symmetry as the open horizon.
Step back and see the arc of this rung. Guide 1 built complex manifolds and the integrability that makes J honest; Guide 2 split forms by type and gave us Dolbeault cohomology; Guide 3 laid metrics down and earned the Kähler identities; Guide 4 turned them into Hodge theory and Hard Lefschetz. This guide cashed all of it in twice over: once for the Kodaira embedding theorem, which makes positivity of a line bundle synonymous with being a projective variety, and once for Calabi-Yau manifolds, where the canonical bundle's triviality yields Ricci-flat metrics by Yau's solution of the Calabi conjecture. From here the road forks toward the algebraic-geometry tracks (sheaves, schemes, Serre duality) and toward symplectic geometry and mirror symmetry — but every fork starts from a closed positive (1,1)-form, the quiet hero of the whole rung.