Dolbeault cohomology
/ dol-BOH /
De Rham cohomology measures the difference between forms that are closed (no boundary) and forms that are exact (already a boundary), using the operator d. Dolbeault cohomology does the same thing but with the operator partial-bar instead of d, and refined by bidegree. It captures the obstruction to solving the del-bar equation partial-bar u = f, and it is the natural cohomology of a complex manifold — the holomorphic analogue of de Rham theory.
Precisely, fix the bidegree p. Because partial-bar^2 = 0, for each p the spaces of (p,q)-forms with q = 0, 1, 2, ... form a complex (the Dolbeault complex), and its cohomology in degree q is H^{p,q}(M) = (partial-bar-closed (p,q)-forms) modulo (partial-bar-exact ones), i.e. ker(partial-bar) / image(partial-bar) at bidegree (p,q). The case p = 0 is special: H^{0,0} is the holomorphic functions, and more generally H^{p,0} is the space of global holomorphic p-forms. By the Dolbeault theorem, H^{p,q}(M) is canonically isomorphic to the sheaf cohomology H^q(M, Omega^p) of the sheaf of holomorphic p-forms — this is the bridge that lets analysts compute with forms and algebraic geometers compute with sheaves and get the same answer.
On a compact complex manifold these groups are finite-dimensional, and their dimensions h^{p,q} are the Hodge numbers, often displayed in the Hodge diamond. On a compact Kähler manifold the Hodge decomposition relates them to ordinary topology: the k-th Betti number is the sum over p+q=k of h^{p,q}, and Hodge symmetry gives h^{p,q} = h^{q,p}. A key honesty point: this clean relationship to de Rham/Betti numbers is special to the Kähler case. On a general compact complex manifold (e.g. a Hopf surface) the Hodge numbers need not sum to the Betti numbers and the symmetry h^{p,q} = h^{q,p} can fail — Dolbeault cohomology is genuinely finer and more delicate than de Rham cohomology there.
For complex projective space CP^n, the Hodge numbers are h^{p,p} = 1 for 0 <= p <= n and h^{p,q} = 0 when p is not equal to q. So the only nonzero Dolbeault groups sit on the diagonal of the Hodge diamond, matching the fact that the even Betti numbers of CP^n are all 1 and the odd ones are 0.
The Hodge diamond of CP^n: a single 1 down the main diagonal, zeros off it.
The tidy link 'Betti number = sum of Hodge numbers, with h^{p,q} = h^{q,p}' is a Kähler theorem, not a general complex-manifold fact. On non-Kähler complex manifolds it can fail outright.