Complex and Kähler geometry; Hodge theory

Take $\C^2 \setminus \{0\}$ and identify each point $z$ with $2z$. The quotient $H = (\C^2 \setminus \{0\}) / (z \sim 2z)$ — a fundamental domain is the shell $1 \le |z| \le 2$ with its two boundary spheres glued — is a compact complex surface — the Hopf surface — and it is diffeomorphic to $S^1 \times S^3$. Its first Betti number is $b_1 = 1$.

No smooth complex projective surface, and more generally no compact Kähler manifold, can have $b_1 = 1$. The reason is one of the great theorems of twentieth-century geometry: on a compact Kähler manifold every cohomology class splits into pieces of type $(p, q)$, and the pieces of types $(p, q)$ and $(q, p)$ are complex conjugates of each other. So $b_1 = h^{1,0} + h^{0,1} = 2h^{1,0}$ is even.

The same Hodge decomposition organises a whole landscape. It explains why algebraic varieties have rigid topology; it gives Calabi–Yau manifolds their Hodge diamonds (the quintic threefold has $h^{2,1} = 101$, a number string theorists and enumerative geometers care about); and it frames the Hodge conjecture, one of the seven Millennium Problems — the question of which cohomology classes come from algebraic subvarieties. This lesson is an orientation to that landscape: the definitions, the linear algebra proved in full, the Hodge theorem stated precisely with its consequences proved, worked Hodge diamonds, and a clear line between theorems and open problems.

On $\C^n = \R^{2n}$ three structures live together:

• the Euclidean metric $g = \sum (dx_j^2 + dy_j^2)$;
• the complex structure $J$, multiplication by $i$, with $J \partial_{x_j} = \partial_{y_j}$;
• the symplectic form $\omega = \sum dx_j \wedge dy_j = \frac{i}{2} \sum dz_j \wedge d\bar z_j$.

They determine each other: $\omega(u, v) = g(Ju, v)$, and $J$ is an isometry. A Kähler manifold is a manifold that carries all three, compatibly, with $J$ coming from holomorphic coordinates and $\omega$ closed. The striking fact is that this is equivalent to asking that $J$ be parallel for the Levi-Civita connection of $g$ — so the complex structure, the metric and the symplectic form are "locked together" as tightly as possible. A Kähler metric osculates the flat metric to second order in suitable holomorphic coordinates: all its curvature appears at second order, never at first.

Why does this constrain topology? Hodge's insight was to represent each de Rham class by its unique harmonic form — the one of least energy in its class. On a Kähler manifold the Laplacian is so well adapted to the complex structure that it does not mix types: the $(p, q)$-parts of a harmonic form are harmonic. Topology (de Rham cohomology) thereby inherits the algebra of types, and complex conjugation exchanges $(p, q)$ with $(q, p)$.

The prerequisites are Riemann surfaces, Riemannian geometry, vector bundles and Chern classes and the language of algebraic geometry.

(Mathlib's inner product is conjugate-linear in the first variable, so the formal version uses $\omega = \operatorname{Im}\langle u, v\rangle$; the two conventions differ by a conjugation.)

Criterion (b) is what the second experiment tests symbolically: the Fubini–Study metric passes, the Hopf metric $|dz|^2/|z|^2$ fails.

This is the same argument as for symplectic manifolds (see symplectic geometry): every Kähler manifold is symplectic. The converse fails — Thurston's example (1976), the Kodaira–Thurston manifold, is a compact symplectic $4$-manifold with $b_1 = 3$, which by the next section cannot be Kähler.

Let $(X, g)$ be compact Kähler of complex dimension $n$. With the formal adjoints $d^*, \partial^*, \bar\partial^*$ for the $L^2$ inner product on forms, define the Laplacians $\Delta_d = dd^* + d^*d$ and $\Delta_{\bar\partial} = \bar\partial\bar\partial^* + \bar\partial^*\bar\partial$, and let $L = \omega \wedge \cdot$ with adjoint $\Lambda$.

The proof is elliptic PDE: $\Delta$ is an elliptic operator, so it has a Green operator on a compact manifold. See Griffiths–Harris Ch. 0 §6, Wells Ch. IV, or Warner's Foundations of Differentiable Manifolds and Lie Groups, Ch. 6.

The identities themselves are first-order and pointwise. Since a Kähler metric agrees with the flat metric to second order at any point in suitable holomorphic coordinates, it suffices to check them on $\C^n$, where they are a computation (Huybrechts, Prop. 3.1.12; Griffiths–Harris, pp. 111–115). The consequences for the Laplacians follow in two lines. First, from $i\bar\partial^* = [\Lambda, \partial]$ and $\partial^2 = 0$,

and its conjugate $\bar\partial\partial^* + \partial^*\bar\partial = 0$ as well, so the mixed terms in $\Delta_d = (\partial + \bar\partial)(\partial^* + \bar\partial^*) + (\partial^* + \bar\partial^*)(\partial + \bar\partial)$ vanish and $\Delta_d = \Delta_\partial + \Delta_{\bar\partial}$. Second, substituting $-i\partial^* = [\Lambda, \bar\partial]$ and $i\bar\partial^* = [\Lambda, \partial]$,

so $\Delta_\partial = \Delta_{\bar\partial}$ and $\Delta_d = 2\Delta_{\bar\partial}$.

Two further theorems complete the classical picture.

Hard Lefschetz (stated). For a compact Kähler manifold, $L^{k} : H^{n-k}(X) \to H^{n+k}(X)$ is an isomorphism for every $k \ge 0$. Consequently $L : H^{j} \to H^{j+2}$ is injective for $j \lt n$, and $b_0 \le b_2 \le b_4 \le \cdots \le b_{2\lfloor n/2 \rfloor}$, similarly for odd degrees. The primitive cohomology carries the Hodge–Riemann bilinear relations, a definiteness property; for surfaces it gives the Hodge index theorem: the intersection form on $H^{1,1}_\R$ has signature $(1, h^{1,1} - 1)$.

The $\partial\bar\partial$-lemma (stated). On a compact Kähler manifold, a $d$-closed form of pure type that is $d$-exact (or $\partial$- or $\bar\partial$-exact) is $\partial\bar\partial$-exact. Deligne, Griffiths, Morgan and Sullivan (1975) deduced that compact Kähler manifolds are formal: their real homotopy type is determined by the cohomology ring, and all Massey products vanish. This is a third, independent topological restriction, used to show that certain symplectic manifolds with even Betti numbers are still not Kähler.

The Ricci form of a Kähler metric, $\rho(u, v) = \operatorname{Ric}(Ju, v)$, is a closed real $(1,1)$-form; locally it is $-i\partial\bar\partial \log\det(g_{j\bar k})$ up to a positive constant fixed by the conventions, and $[\rho] = 2\pi c_1(X)$ in $H^2(X; \R)$ whatever the metric. So $c_1(X)$ is the only obstruction to Ricci-flatness within cohomology — and Calabi asked (1954–57) whether it is the only obstruction at all.

Theorem (Yau, 1976–78; the Calabi conjecture). Let $X$ be compact Kähler with Kähler form $\omega$, and let $\rho'$ be any closed real $(1,1)$-form representing $2\pi c_1(X)$. Then there is a unique Kähler form $\omega'$ in the class $[\omega]$ whose Ricci form is $\rho'$. In particular, if $c_1(X) = 0$ in $H^2(X; \R)$, each Kähler class contains a unique Ricci-flat Kähler metric.

Writing $\omega' = \omega + i\partial\bar\partial\varphi$ turns the problem into the complex Monge–Ampère equation $(\omega + i\partial\bar\partial\varphi)^n = e^{F}\omega^n$ with $\int e^F\omega^n = \int \omega^n$. Calabi proved uniqueness; Yau proved existence by the continuity method, the decisive step being an a priori $C^0$ estimate for $\varphi$. Ricci-flat Kähler metrics have holonomy in $SU(n)$; K3 surfaces and the quintic threefold carry them, although no explicit formula is known for any compact example other than flat tori and their quotients.

Kähler–Einstein metrics ($\rho = \lambda\omega$) exist when $c_1(X) \lt 0$ (Aubin and Yau, 1976–78) and when $c_1(X) = 0$ (Yau). When $c_1(X) \gt 0$ (Fano manifolds) there are obstructions (Matsushima, Futaki), and the Yau–Tian–Donaldson conjecture — existence if and only if $X$ is K-polystable — was proved for Fano manifolds by Chen, Donaldson and Sun (2015).

The Hodge diamond of a Calabi–Yau threefold is determined by two numbers, $h^{1,1}$ and $h^{2,1}$: for the quintic, $h^{1,1} = 1$ (the hyperplane class, by the Lefschetz hyperplane theorem) and $h^{2,1} = 101$ (the number of independent deformations of the complex structure, by the Bogomolov–Tian–Todorov theorem and $H^1(T_X) \cong H^{2,1}$). The experiment computes $101$ by counting monomials. Mirror symmetry predicts a "mirror quintic" with these two numbers exchanged; it exists (Greene–Plesser, 1990).

Let $X$ be a smooth complex projective variety of dimension $n$. A Hodge class of degree $2p$ is an element of

If $Z \subset X$ is a subvariety of codimension $p$, its fundamental class $[Z] \in H^{2p}(X; \Q)$ is a Hodge class: integrating a closed $(r, s)$-form with $r + s = 2(n-p)$ over the complex $(n-p)$-dimensional $Z$ gives zero unless $(r, s) = (n-p, n-p)$, so the Poincaré dual of $[Z]$ pairs to zero with every $H^{r,s}$ except $H^{n-p,n-p}$, which places it in $H^{p,p}$.

The Hodge conjecture (Hodge, 1950, in the corrected rational form). Every Hodge class on a smooth complex projective variety is a $\Q$-linear combination of fundamental classes of algebraic subvarieties. It is one of the Clay Mathematics Institute's seven Millennium Problems (2000).

What is known:

• $p = 1$: true. The Lefschetz $(1,1)$-theorem (Lefschetz, 1924; sheaf-theoretic proof by Kodaira and Spencer, 1953): an integral class in $H^2(X; \Z)$ is the first Chern class of a holomorphic line bundle if and only if its image in $H^2(X; \C)$ has type $(1,1)$. Proof sketch: the exponential sequence $0 \to \Z \to \mathcal{O}_X \to \mathcal{O}_X^* \to 0$ gives $H^1(X, \mathcal{O}^*) \xrightarrow{c_1} H^2(X; \Z) \to H^2(X, \mathcal{O}_X) = H^{0,2}$, and a real class maps to zero in $H^{0,2}$ exactly when, by Hodge symmetry, its $(2,0)$ and $(0,2)$ parts vanish. On a projective variety every line bundle is $\mathcal{O}(D)$ for a divisor $D$, so $c_1 = [D]$ is algebraic.
• $p = n - 1$: true, by hard Lefschetz from the case $p = 1$.
• Many special cases are known — for instance for abelian varieties of dimension at most $3$, where it follows from the cases $p = 1$ and $p = n - 1$, and for various varieties with large symmetry. The conjecture is open in general, even for abelian varieties of higher dimension.

What is false — the conjecture is stated carefully for good reasons:

• Integral coefficients. Atiyah and Hirzebruch (1962) found torsion integral Hodge classes that are not algebraic; Kollár (1992) found non-torsion examples. The integral Hodge conjecture is false.
• Kähler manifolds. Voisin (2002) constructed compact Kähler manifolds whose Hodge classes are not even combinations of Chern classes of coherent sheaves; so the natural Kähler generalisation is false, and projectivity is essential.
• Kodaira's problem. Kodaira showed (1954) that a compact Kähler manifold with an integral Kähler class is projective, and (1963) that every compact Kähler surface deforms to a projective one. Voisin (2004) showed that in dimension at least $4$ there are compact Kähler manifolds not even homotopy equivalent to any projective manifold.

Mathlib has complex manifolds (manifolds modelled on $\C^n$ with holomorphic transition maps) and inner product spaces, but no Kähler manifolds, Dolbeault cohomology or Hodge theory yet. Formalised here: the pointwise linear algebra of Proposition 1, the arithmetic of Corollary 7 (ii) and (iv), the Hodge numbers of tori, and the rigidity of holomorphic functions.

Mathlib's complex inner product inner ℂ u v is conjugate-linear in u, so the Kähler form is its imaginary part:

/-- The Riemannian metric `g(u, v) = Re h(u, v)` underlying a Hermitian inner product `h`. -/
noncomputable def metric (u v : E) : ℝ := (inner ℂ u v).re

/-- The fundamental (Kähler) form `ω(u, v) = Im h(u, v)` of a Hermitian inner product `h`. On `ℂ`
with `h(u, v) = conj(u) v` it is `dx ∧ dy`: `ω(1, i) = Im(i) = 1`. -/
noncomputable def kahlerForm (u v : E) : ℝ := (inner ℂ u v).im

/-- **The triple is compatible**: `ω(u, v) = g(Ju, v)` with `J = i·`. Proof:
`h(iu, v) = conj(i) h(u, v) = -i h(u, v)`, and `Re(-i z) = Im z`. -/
theorem kahler_form_eq_metric_J (u v : E) :
    kahlerForm u v = metric (Complex.I • u) v := by
  simp [kahlerForm, metric, inner_smul_left]

/-- The complex structure is an isometry: `g(Ju, Jv) = g(u, v)`, because
`h(iu, iv) = conj(i) · i · h(u, v) = h(u, v)`. -/
theorem metric_J_invariant (u v : E) :
    metric (Complex.I • u) (Complex.I • v) = metric u v := by
  simp [metric, inner_smul_left, inner_smul_right, ← mul_assoc]

/-- The Kähler form is `J`-invariant: `ω(Ju, Jv) = ω(u, v)`. So `ω` has type `(1, 1)`. -/
theorem kahler_form_J_invariant (u v : E) :
    kahlerForm (Complex.I • u) (Complex.I • v) = kahlerForm u v := by
  simp [kahlerForm, inner_smul_left, inner_smul_right, ← mul_assoc]

/-- The Kähler form is skew: `ω(v, u) = -ω(u, v)`, because `h(v, u) = conj h(u, v)` and
conjugation negates imaginary parts. -/
theorem kahler_form_antisymm (u v : E) : kahlerForm v u = -kahlerForm u v := by
  rw [kahlerForm, kahlerForm, ← inner_conj_symm u v, Complex.conj_im, neg_neg]

/-- The Kähler form is alternating: `ω(u, u) = 0`. (Solution of the Lean exercise: it follows
from skew-symmetry, since a real number equal to its negative is zero.) -/
theorem kahler_form_self (u : E) : kahlerForm u u = 0 := by
  have h := kahler_form_antisymm u u
  linarith

/-- **Taming**: `ω(u, Ju) = |u|²`. The Kähler form is positive on every complex line, which is
why complex submanifolds of a Kähler manifold are volume-minimising and have positive volume. -/
theorem kahler_form_J_self (u : E) : kahlerForm u (Complex.I • u) = ‖u‖ ^ 2 := by
  simp [kahlerForm, inner_smul_right, inner_self_eq_norm_sq_to_K]
  norm_cast

/-- The Kähler form is non-degenerate: if `ω(u, v) = 0` for all `v`, then
`|u|² = ω(u, Ju) = 0`. With closedness (`dω = 0`, the Kähler condition) this makes every Kähler
manifold symplectic. -/
theorem kahler_form_nondegenerate {u : E} (h : ∀ v, kahlerForm u v = 0) : u = 0 := by
  have := h (Complex.I • u)
  rw [kahler_form_J_self] at this
  simpa using this

From Hodge symmetry to topology: the parity argument, and the Hopf surface.

/-- **The arithmetic behind "odd Betti numbers are even".** If `h p q = h q p` (Hodge symmetry),
then for odd `k = 2m + 1` the sum `b_k = ∑_{p + q = k} h^{p,q}` is even: split the sum at the middle
and reflect the second half, `p ↦ k - p`, which has no fixed point when `k` is odd. On a compact
Kähler manifold the Hodge decomposition gives `b_k = ∑_{p+q=k} h^{p,q}`; that analytic input is not
formalised. -/
theorem odd_betti_even {h : ℕ → ℕ → ℕ} (hs : ∀ p q, h p q = h q p) (m : ℕ) :
    Even (∑ p ∈ range (2 * m + 2), h p (2 * m + 1 - p)) := by
  set f : ℕ → ℕ := fun p => h p (2 * m + 1 - p) with hf
  have hsplit : ∑ p ∈ range (2 * m + 2), f p =
      ∑ p ∈ range (m + 1), f p + ∑ p ∈ range (m + 1), f (m + 1 + p) := by
    rw [show 2 * m + 2 = (m + 1) + (m + 1) by omega, Finset.sum_range_add]
  have hrefl : ∑ p ∈ range (m + 1), f (m + 1 + p) = ∑ p ∈ range (m + 1), f p := by
    rw [← Finset.sum_range_reflect]
    apply Finset.sum_congr rfl
    intro p hp
    have hp' : p < m + 1 := mem_range.mp hp
    simp only [hf]
    rw [hs]
    congr 1 <;> omega
  rw [hsplit, hrefl]
  exact ⟨_, rfl⟩

/-- **Why the Hopf surface is not Kähler.** With Hodge symmetry, `b₁ = h^{1,0} + h^{0,1}` is twice
`h^{1,0}`, so it can never be `1`. The Hopf surface `S¹ × S³` has `b₁ = 1`. -/
theorem hopf_obstruction {h : ℕ → ℕ → ℕ} (hs : ∀ p q, h p q = h q p) :
    h 1 0 + h 0 1 ≠ 1 := by
  rw [hs 1 0]
  omega

The Hodge numbers of a complex torus add up to the Betti numbers of the real torus — Vandermonde's identity, from Mathlib:

/-- **Hodge numbers of a complex torus.** For `X = ℂⁿ/Λ`, `h^{p,q} = C(n, p) C(n, q)` (translation-
invariant forms `dz_I ∧ dz̄_J`), and they add up to the Betti numbers `b_k = C(2n, k)` of the real
torus `T^{2n}`: `C(n + n, k) = ∑_{p + q = k} C(n, p) C(n, q)`. This applies Mathlib's Vandermonde
identity `Nat.add_choose_eq`. -/
theorem torus_betti (n k : ℕ) :
    (n + n).choose k = ∑ pq ∈ antidiagonal k, n.choose pq.1 * n.choose pq.2 :=
  Nat.add_choose_eq n n k

And the maximum principle on compact complex manifolds, which Mathlib proves for manifolds modelled on any boundaryless model with corners over $\C$:

/-- **Holomorphic functions on a compact connected complex manifold are constant** (maximum
modulus principle). This is why complex geometry studies sections of line bundles, sheaves and
forms instead of functions. It applies Mathlib's `MDifferentiable.apply_eq_of_compactSpace`;
`MDifferentiable I 𝓘(ℂ, F) f` is holomorphy for a manifold modelled on `I`. -/
theorem holomorphic_const_of_compact {E' : Type*} [NormedAddCommGroup E'] [NormedSpace ℂ E']
    {F : Type*} [NormedAddCommGroup F] [NormedSpace ℂ F]
    {H : Type*} [TopologicalSpace H] {I : ModelWithCorners ℂ E' H} [I.Boundaryless]
    {M : Type*} [TopologicalSpace M] [ChartedSpace H M] [IsManifold I 1 M]
    [CompactSpace M] [PreconnectedSpace M] {f : M → F}
    (hf : MDifferentiable I (modelWithCornersSelf ℂ F) f) (a b : M) : f a = f b :=
  hf.apply_eq_of_compactSpace a b

Not formalised: the Dolbeault complex and Dolbeault's theorem, the Hodge theorem (which needs elliptic regularity on manifolds), the Kähler identities and the Hodge decomposition itself — the declarations above take Hodge symmetry as a hypothesis on an abstract array of numbers — hard Lefschetz, Yau's theorem, and any statement about algebraic cycles.

Griffiths' theorem (1969) computes the primitive middle cohomology of a smooth hypersurface $X = \{F = 0\} \subset \mathbb{CP}^{n+1}$ of degree $d$ from its Jacobian ring $R = \C[x_0, \dots, x_{n+1}]/(\partial F/\partial x_i)$: $h^{n-q,q}_{\mathrm{prim}}(X) = \dim R_{(q+1)d - n - 2}$. For the Fermat polynomial $F = \sum x_i^d$ the Jacobian ideal is $(x_i^{d-1})$, so $R$ has a basis of monomials with all exponents at most $d - 2$, and the Hodge numbers become a counting problem. The first experiment counts, assembles the diamonds, and checks them against the Euler characteristic computed independently from Chern classes, $c(T_X) = (1 + H)^{n+2}/(1 + dH)$:

"""Hodge numbers of hypersurfaces by counting monomials.

For a smooth hypersurface X of degree d in complex projective space P^(n+1)
(so dim X = n), Griffiths' theorem identifies the primitive middle Hodge
numbers with graded pieces of the Jacobian ring C[x_0..x_(n+1)]/(dF/dx_i).
For the Fermat polynomial F = sum x_i^d the Jacobian ideal is (x_i^(d-1)),
so a basis is given by monomials with every exponent at most d - 2, and

    h^(n-q, q)_prim = #{monomials of degree (q+1)d - (n+2), exponents <= d-2}.

We count them exactly, assemble the Hodge diamond (the other entries come
from the Lefschetz hyperplane theorem), and check it against an independent
computation of the Euler characteristic from Chern classes:
chi(X) = d * [H^n] (1 + H)^(n+2) / (1 + d H).

Evidence: exact integer computation of numbers that theorems (Griffiths,
Lefschetz, Chern-Gauss-Bonnet) identify with Hodge numbers and Euler
characteristics. The agreement of the two formulas is a consistency check.
"""

from math import comb

def bounded_compositions(total, parts, cap):
    """Number of (a_1..a_parts) with 0 <= a_i <= cap and sum total."""
    if total < 0:
        return 0
    ways = [1] + [0] * total
    for _ in range(parts):
        new = [0] * (total + 1)
        for s, w in enumerate(ways):
            if w:
                for a in range(min(cap, total - s) + 1):
                    new[s + a] += w
        ways = new
    return ways[total]

def primitive_middle(n, d):
    """[h^(n,0)_prim, h^(n-1,1)_prim, ..., h^(0,n)_prim] for degree d in P^(n+1)."""
    return [bounded_compositions((q + 1) * d - (n + 2), n + 2, d - 2)
            for q in range(n + 1)]

def hodge_diamond(n, d):
    """Dict (p, q) -> h^(p,q) for a smooth degree-d hypersurface of dim n."""
    h = {(p, q): 0 for p in range(n + 1) for q in range(n + 1)}
    for p in range(n + 1):
        if 2 * p != n:
            h[(p, p)] = 1  # Lefschetz: powers of the hyperplane class
    for q, value in enumerate(primitive_middle(n, d)):
        h[(n - q, q)] += value
    if n % 2 == 0:
        h[(n // 2, n // 2)] += 1  # the hyperplane class in the middle
    return h

def betti(h, n):
    """Betti numbers b_k = sum_{p+q=k} h^(p,q)."""
    return [sum(h[(p, k - p)] for p in range(n + 1) if 0 <= k - p <= n)
            for k in range(2 * n + 1)]

def euler_from_hodge(h):
    """chi = sum (-1)^(p+q) h^(p,q)."""
    return sum((-1) ** (p + q) * v for (p, q), v in h.items())

def euler_from_chern(n, d):
    """chi = d * coefficient of H^n in (1 + H)^(n+2) / (1 + dH)."""
    return d * sum(comb(n + 2, k) * (-d) ** (n - k) for k in range(n + 1))

def symmetric(h, n):
    """Hodge symmetry h^(p,q) = h^(q,p) and Serre duality h^(p,q) = h^(n-p,n-q)."""
    return all(h[(p, q)] == h[(q, p)] == h[(n - p, n - q)] for p, q in h)

def diamond_lines(h, n, width=5):
    """The Hodge diamond as centred text lines, h^(n,n) at the top."""
    lines = []
    for k in range(2 * n, -1, -1):
        row = [h[(p, k - p)] for p in range(n, -1, -1) if 0 <= k - p <= n]
        text = "".join(f"{v:^{width}}" for v in row)
        lines.append(text.center(width * (n + 1)).rstrip())
    return lines

CASES = [
    ("plane cubic (elliptic curve)", 1, 3),
    ("plane quartic curve", 1, 4),
    ("quartic surface (K3)", 2, 4),
    ("quintic surface", 2, 5),
    ("cubic threefold", 3, 3),
    ("quintic threefold (Calabi-Yau)", 3, 5),
]

def main():
    print(f"{'hypersurface':<31}{'n':>2}{'d':>3}  {'middle h^(n-q,q)':<17}"
          f"{'chi Hodge':>10}{'chi Chern':>10}")
    for name, n, d in CASES:
        h = hodge_diamond(n, d)
        middle = ",".join(str(h[(n - q, q)]) for q in range(n + 1))
        print(f"{name:<31}{n:>2}{d:>3}  {middle:<17}"
              f"{euler_from_hodge(h):>10}{euler_from_chern(n, d):>10}")
        assert symmetric(h, n)
        assert all(b % 2 == 0 for b in betti(h, n)[1::2])
    print("All diamonds satisfy h^(p,q) = h^(q,p) = h^(n-p,n-q), and every")
    print("odd Betti number is even.")
    for title, n, d in [("Quartic K3 surface:", 2, 4),
                        ("Quintic threefold:", 3, 5)]:
        h = hodge_diamond(n, d)
        print()
        print(f"{title} Betti numbers {betti(h, n)}")
        for line in diamond_lines(h, n):
            print("    " + line)
    print()
    print("Epistemic status: exact integer counts. That they are Hodge numbers")
    print("is Griffiths' theorem on the Jacobian ring and the Lefschetz")
    print("hyperplane theorem, cited not proved here; the match with the")
    print("Chern-class Euler characteristic is an independent consistency check.")

if __name__ == "__main__":
    main()

The plane curves recover the genus formula $g = \frac{(d-1)(d-2)}{2}$; the quartic surface gives $1, 20, 1$; the quintic threefold gives $h^{2,1} = 101$ and $\chi = 2(h^{1,1} - h^{2,1}) = -200$.

The second experiment tests Proposition 2's criterion $\partial_l g_{j\bar k} = \partial_j g_{l\bar k}$ with SymPy, treating $z$ and $\bar z$ as independent variables (Wirtinger calculus). The flat and Fubini–Study metrics pass; the scale-invariant Hopf metric, which descends to $S^1 \times S^3$, fails:

"""Is a Hermitian metric Kahler? A symbolic check in coordinates.

A Hermitian metric on an open set of C^n is a positive definite Hermitian
matrix of functions g_{i jbar}; its fundamental form is
omega = (i/2) sum g_{i jbar} dz_i ^ dzbar_j, and omega is closed (the metric
is Kahler) exactly when d_k g_{i jbar} = d_i g_{k jbar} for all i, j, k
(together with the conjugate equations, which follow by symmetry).

We treat z and zbar as independent symbols, as in Wirtinger calculus, and
test three metrics on C^2: the flat metric, the Fubini-Study metric
g = d dbar log(1 + |z|^2) of the affine chart of CP^2 (Kahler, with Kahler
potential log(1 + |z|^2)), and the Hopf metric g = delta / |z|^2 on
C^2 minus 0, which is invariant under z -> 2z and so descends to the Hopf
surface S^1 x S^3, but is not Kahler. Finally SymPy integrates the
Fubini-Study area form of CP^1.

Evidence: symbolic computation with SymPy (exact, but relying on SymPy's
implementation of differentiation, simplification and integration).
"""

import sympy as sp

Z = sp.symbols("z1 z2")
ZB = sp.symbols("zb1 zb2")

def from_potential(phi):
    """g_{i jbar} = d^2 phi / dz_i dzbar_j for a Kahler potential phi."""
    return sp.Matrix(2, 2, lambda i, j: sp.diff(phi, Z[i], ZB[j]))

def kahler_defects(g):
    """All nonzero d_k g_{i jbar} - d_i g_{k jbar}, simplified."""
    out = []
    for i in range(2):
        for j in range(2):
            for k in range(2):
                e = sp.simplify(sp.diff(g[i, j], Z[k]) - sp.diff(g[k, j], Z[i]))
                if e != 0:
                    out.append(((i + 1, j + 1, k + 1), e))
    return out

def is_hermitian_symbolically(g):
    """g_{j ibar} is the conjugate of g_{i jbar} (swapping z and zbar)."""
    swap = {**{Z[i]: ZB[i] for i in range(2)}, **{ZB[i]: Z[i] for i in range(2)}}
    return all(sp.simplify(g[j, i] - g[i, j].xreplace(swap)) == 0
               for i in range(2) for j in range(2))

def scale_invariant(g, factor=2):
    """Is g dz dzbar unchanged by z -> factor * z (g rescales by factor^-2)?"""
    sub = {**{Z[i]: factor * Z[i] for i in range(2)},
           **{ZB[i]: factor * ZB[i] for i in range(2)}}
    return all(sp.simplify(g[i, j].xreplace(sub) * factor ** 2 - g[i, j]) == 0
               for i in range(2) for j in range(2))

def fubini_study_area_cp1():
    """Integral over C of the Fubini-Study area density 1/(1 + r^2)^2."""
    r, t = sp.symbols("r t", positive=True)
    density = 1 / (1 + r ** 2) ** 2
    return sp.integrate(sp.integrate(density * r, (r, 0, sp.oo)), (t, 0, 2 * sp.pi))

def metrics():
    """The three test metrics on (an open subset of) C^2."""
    norm2 = Z[0] * ZB[0] + Z[1] * ZB[1]
    return [
        ("flat", sp.eye(2)),
        ("Fubini-Study", from_potential(sp.log(1 + norm2))),
        ("Hopf |dz|^2/|z|^2", sp.eye(2) / norm2),
    ]

def main():
    print(f"SymPy {sp.__version__}; z and zbar treated as independent symbols.")
    for name, g in metrics():
        defects = kahler_defects(g)
        print(f"{name}:")
        print(f"  Hermitian: {is_hermitian_symbolically(g)};"
              f" invariant under z -> 2z: {scale_invariant(g)}")
        if defects:
            (i, j, k), e = defects[0]
            print(f"  NOT Kahler: {len(defects)} nonzero defects, e.g.")
            print(f"  d_{k} g_{i}{j}bar - d_{i} g_{k}{j}bar = {e}")
        else:
            print("  Kahler: d_k g_(i jbar) = d_i g_(k jbar) for all i, j, k")
    fs = from_potential(sp.log(1 + Z[0] * ZB[0] + Z[1] * ZB[1]))
    print("Fubini-Study g_(1 1bar) =", sp.factor(fs[0, 0]))
    print("Fubini-Study area of CP^1 =", fubini_study_area_cp1())
    print()
    print("Epistemic status: exact symbolic computation, as reliable as")
    print("SymPy's differentiation, simplification and integration. A zero")
    print("defect is an identity of rational functions; it shows these")
    print("particular metrics are (or are not) Kahler, nothing more.")

if __name__ == "__main__":
    main()