Hodge symmetry
/ HODGE rhymes with 'dodge' /
Ordinary topology assigns to a space its Betti numbers b_k, counting independent k-dimensional holes. On a compact Kähler manifold these holes carry extra structure: each one can be further sorted by how 'holomorphic' versus 'antiholomorphic' it is, giving a finer set of numbers called Hodge numbers h^{p,q}. Hodge symmetry is the elegant statement that this refined table has a built-in mirror symmetry — swapping the holomorphic and antiholomorphic counts leaves it unchanged.
Precisely, on a compact Kähler manifold the de Rham cohomology with complex coefficients decomposes as a direct sum H^k(M, C) = sum over p+q=k of H^{p,q}(M), where H^{p,q} is the Dolbeault cohomology of bidegree (p,q); the dimension of H^{p,q} is the Hodge number h^{p,q}. Hodge symmetry asserts two things: first that this decomposition exists at all (the Hodge decomposition), and second that complex conjugation gives an isomorphism H^{p,q} isomorphic to the conjugate of H^{q,p}, so h^{p,q} = h^{q,p}. A consequence is that the k-th Betti number is the sum over p+q=k of h^{p,q}, and another, combining with Serre duality, is the full diamond symmetry h^{p,q} = h^{q,p} = h^{n-p,n-q}. A famous corollary: every odd Betti number b_{2m+1} is even, since it is twice the sum of h^{p,q} over p<q with p+q odd.
This is one of the sharpest dividing lines between Kähler and non-Kähler complex geometry. The proof rests entirely on the Kähler identities, which make the d-Laplacian and the partial-bar-Laplacian agree so that harmonic forms split by type. The honesty caveat is essential: Hodge symmetry can fail on a compact complex manifold that is not Kähler — the Hopf surface has b_1 = 1 (odd), which is flatly impossible for a Kähler manifold, and its Hodge numbers are not symmetric. So whenever you invoke h^{p,q} = h^{q,p} you are silently using the Kähler hypothesis.
For a compact Riemann surface of genus g, the Hodge diamond in degree 1 reads h^{1,0} = h^{0,1} = g, and Hodge symmetry h^{1,0} = h^{0,1} just says the holomorphic 1-forms and their conjugates have the same dimension g. The Betti number b_1 = h^{1,0} + h^{0,1} = 2g is even, as it must be for any Kähler manifold.
On a genus-g curve, h^{1,0} = h^{0,1} = g, forcing b_1 = 2g to be even.
Hodge symmetry is a Kähler theorem, not a general complex one. The Hopf surface (b_1 = 1, odd) violates it outright, certifying that being a complex manifold is not enough — you genuinely need d omega = 0.