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The Lumenary · Research pathways

Complex and Kähler geometry; Hodge theory

Name: ______________________

Date: ______________________

You will learn to

Define complex manifolds, Hermitian and Kähler metrics, and the Dolbeault complex, with projective varieties, tori and the Fubini–Study metric as the main examples
Prove the linear algebra of the compatible triple $(g, J, \omega)$ and the local criterion $\partial_k g_{i\bar\jmath} = \partial_i g_{k\bar\jmath}$ for a metric to be Kähler
State the Hodge decomposition and Hodge symmetry, and deduce topological restrictions on compact Kähler manifolds (odd Betti numbers even, hard Lefschetz inequalities, the Hopf surface is not Kähler)
Compute Hodge diamonds of curves, tori, K3 surfaces and the quintic threefold
State the Calabi conjecture (Yau's theorem) and the Hodge conjecture, separating theorems from open problems

1.
A K3 surface has $b_1 = 0$ , $b_2 = 22$ and $h^{2,0} = 1$ . Use the Hodge decomposition and Hodge symmetry to find $h^{1,1}$ .
2.
The quintic threefold $X \subset \mathbb{CP}^4$ is a Calabi–Yau threefold with $h^{3,0} = 1$ and $h^{2,1} = 101$ . What is its third Betti number $b_3$ ?
3.
Which of these compact complex manifolds admit a Kähler metric? (Choose all that apply.)
- A. $\mathbb{CP}^n$
- B.Any smooth complex projective variety
- C.Any complex torus $\C^n/\Lambda$ , projective or not
- D.Any compact Riemann surface
- E.The Hopf surface $(\C^2 \setminus 0)/(z \sim 2z)$
- F.A primary Kodaira surface (a compact complex surface with $b_1 = 3$ )
4.
The proof that the odd Betti numbers of a compact Kähler manifold are even uses $b_k = \sum_{p+q=k} h^{p,q}$ and one more fact. Which one?
- A.Hodge symmetry $h^{p,q} = h^{q,p}$
- B.Serre duality $h^{p,q} = h^{n-p,n-q}$
- C.Poincaré duality $b_k = b_{2n-k}$
- D.Hard Lefschetz: $b_k \le b_{k+2}$ for $k \lt n$
5.
Which statement is the Hodge conjecture (as it stands today, open)?
- A.On a smooth complex projective variety, every class in $H^{2p}(X; \Q) \cap H^{p,p}(X)$ is a $\Q$ -linear combination of classes of algebraic subvarieties.
- B.On a smooth complex projective variety, every class in $H^{2p}(X; \Z)$ whose image lies in $H^{p,p}(X)$ is a $\Z$ -linear combination of classes of algebraic subvarieties.
- C.On a compact Kähler manifold, every rational Hodge class is a combination of Chern classes of coherent sheaves.
- D.Every class in $H^{p,p}(X)$ (with complex coefficients) is algebraic.
6.
Let $X$ be a compact Kähler manifold with $c_1(X) = 0$ in $H^2(X; \R)$ . What does Yau's theorem (the Calabi conjecture) give?
- A.A unique Ricci-flat Kähler metric in each Kähler class
- B.A flat Kähler metric
- C.A Kähler–Einstein metric with positive Einstein constant
- D.That the canonical bundle is trivial
7.
Which Lean statement correctly formalises "the Kähler form kahlerForm is non-degenerate" (for u : E)?
- A.(∀ v, kahlerForm u v = 0) → u = 0
- B.∀ v, kahlerForm u v = 0 → u = 0
- C.kahlerForm u u = 0 → u = 0
- D.∃ v, kahlerForm u v ≠ 0
8.
By the Lefschetz $(1,1)$ -theorem, the classes of divisors on a projective surface span $H^2(X;\Q) \cap H^{1,1}(X)$ . What is the largest possible Picard number (the rank of this space) for a complex projective K3 surface?
9.
Which of these could not be the list of even Betti numbers $(b_0, b_2, b_4, b_6, b_8)$ of a compact Kähler manifold of complex dimension $4$ ?
- A. $(1, 3, 2, 3, 1)$
- B. $(1, 1, 1, 1, 1)$
- C. $(1, 2, 3, 2, 1)$
- D. $(1, 4, 6, 4, 1)$
10.
For a complex torus of dimension $n$ , $h^{p,q} = \binom{n}{p}\binom{n}{q}$ . Write torus_hodge(n) returning a dictionary {(p, q): h} for $0 \le p, q \le n$ , and betti(h, n) returning the list $[b_0, \dots, b_{2n}]$ with $b_k = \sum_{p+q=k} h^{p,q}$ . The tests check the Betti numbers $\binom{2n}{k}$ of the real torus $T^{2n}$ and Hodge symmetry. (math.comb is available.)
Computational — Practice

Answer key — Complex and Kähler geometry; Hodge theory

1.
$22 = b_2 = h^{2,0} + h^{1,1} + h^{0,2} = 1 + h^{1,1} + 1$ , so $h^{1,1} = 20$ . The quartic surface in $\mathbb{CP}^3$ is an example; the first experiment recovers $1, 20, 1$ by counting monomials.
2.
$b_3 = h^{3,0} + h^{2,1} + h^{1,2} + h^{0,3} = 1 + 101 + 101 + 1 = 204$ . With $b_0 = b_2 = b_4 = b_6 = 1$ and $b_1 = b_5 = 0$ , the Euler characteristic is $4 - 204 = -200$ , matching the Chern-class computation.
3.
Kähler: $\mathbb{CP}^n$ (Fubini–Study), smooth projective varieties (restriction), all complex tori (flat metric), all compact Riemann surfaces (every Hermitian metric is Kähler in complex dimension one). Not Kähler: the Hopf surface ( $b_1 = 1$ ) and a primary Kodaira surface ( $b_1 = 3$ ), because odd Betti numbers of compact Kähler manifolds are even.
4.
For odd $k = 2m + 1$ , $b_k = \sum_{p=0}^{k} h^{p,k-p}$ , and the involution $p \mapsto k - p$ has no fixed point. Hodge symmetry $h^{p,q} = h^{q,p}$ then gives $b_k = 2\sum_{p=0}^{m} h^{p,k-p}$ . Serre and Poincaré duality relate different degrees, and hard Lefschetz gives inequalities. The Lean declaration odd_betti_even formalises exactly this step.
5.
The Hodge conjecture: on a smooth complex projective variety, rational Hodge classes $H^{2p}(X; \Q) \cap H^{p,p}(X)$ are $\Q$ -linear combinations of fundamental classes of algebraic subvarieties. It is known for $p = 1$ (Lefschetz $(1,1)$ -theorem) and $p = n-1$ . The integral version and the Kähler version are false, and complex classes are not even rational in general.
6.
Yau proved that any closed real $(1,1)$ -form representing $2\pi c_1(X)$ is the Ricci form of a unique Kähler metric in each Kähler class. If $c_1(X) = 0$ in $H^2(X;\R)$ we may take the form $0$ , giving a unique Ricci-flat Kähler metric in each class. Flatness is far stronger, and positive Kähler–Einstein metrics need $c_1 \gt 0$ .
7.
Non-degeneracy says: if ω(u, v) = 0 for all v, then u = 0, i.e. (∀ v, kahlerForm u v = 0) → u = 0. Option (b) has the quantifier in the wrong place and is false (take v = u); (c) is false because ω is alternating; (d) is false for u = 0. The Lean proof takes v = J u and uses ω(u, Ju) = |u|².
8.
Divisor classes are rational classes of type $(1,1)$ , so the Picard number is at most $h^{1,1} = 20$ . The bound is attained: the Fermat quartic has Picard number $20$ (such K3 surfaces are called singular K3 surfaces, a historical misnomer). In characteristic $p$ the bound $22$ can be attained, by supersingular K3 surfaces — a phenomenon with no complex analogue.
9.
For $n = 4$ , hard Lefschetz says $L^2 : H^2 \to H^6$ is an isomorphism; since it factors as $H^2 \xrightarrow{L} H^4 \xrightarrow{L} H^6$ , the first map is injective and $b_2 \le b_4$ . The list $(1, 3, 2, 3, 1)$ violates this. The other lists are realised by $\mathbb{CP}^4$ , $\mathbb{CP}^2 \times \mathbb{CP}^2$ and $(\mathbb{CP}^1)^4$ .
10.
```
from math import comb


def torus_hodge(n):
    return {(p, q): comb(n, p) * comb(n, q)
            for p in range(n + 1) for q in range(n + 1)}


def betti(h, n):
    return [sum(h[(p, k - p)] for p in range(n + 1) if 0 <= k - p <= n)
            for k in range(2 * n + 1)]
```
The Betti numbers come out as $\binom{2n}{k}$ by Vandermonde's identity $\sum_{p+q=k}\binom{n}{p}\binom{n}{q} = \binom{2n}{k}$ (the Lean declaration torus_betti), and the Euler characteristic is $(1-1)^{2n} = 0$ .