Complex and Kähler geometry; Hodge theory

A K3 surface has $b_1 = 0$, $b_2 = 22$ and $h^{2,0} = 1$. Use the Hodge decomposition and Hodge symmetry to find $h^{1,1}$.

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 $b_3$?

Which of these compact complex manifolds admit a Kähler metric? (Choose all that apply.)

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?

Which statement is the Hodge conjecture (as it stands today, open)?

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?

Which Lean statement correctly formalises "the Kähler form kahlerForm is non-degenerate" (for u : E)?

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?

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 $4$?

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.)