Pure Hodge Theory

H^n(X,ℂ) = ⊕_{p+q=n} H^{p,q}(X) — the Hodge decomposition of Kähler manifold cohomology

Kähler manifold X

Hodge Numbers h^{p,q}

Symmetries: h^{p,q} = h^{q,p} (complex conj.) = h^{n-p,n-q} (Poincaré)
Hodge theorem:
H^{p,q} = ker(Δ̄) ∩ Ω^{p,q}(X)
(harmonic (p,q)-forms)

Hard Lefschetz:
L^k: H^{n-k} → H^{n+k} is iso
(L = cup product with ω_Kähler)

Hodge conjecture:
H^{p,p}∩H^{2p}(X,ℚ) spanned by algebraic cycles — OPEN (Millennium Problem)