the Dolbeault operators
/ dol-BOH /
On a complex manifold, the ordinary exterior derivative d can be split into the two halves that respect the complex structure: one half differentiates in the holomorphic directions and the other in the antiholomorphic directions. These two halves are the Dolbeault operators, written partial (del) and partial-bar (del-bar). The second of them, partial-bar, is the star of complex geometry: a function is holomorphic exactly when partial-bar f = 0, so partial-bar is the manifold-wide generalization of the Cauchy-Riemann equations packaged as a single operator.
Precisely, using the (p,q)-type decomposition, the exterior derivative on a complex manifold factors as d = partial + partial-bar, where partial sends a (p,q)-form to a (p+1,q)-form and partial-bar sends it to a (p,q+1)-form. In local holomorphic coordinates, partial f = sum (df/dz^i) dz^i and partial-bar f = sum (df/d(z-bar)^j) d(z-bar)^j. Because d^2 = 0 and the bidegrees of partial^2, partial partial-bar + partial-bar partial, and partial-bar^2 are all distinct, each piece vanishes separately: partial^2 = 0, partial-bar^2 = 0, and partial partial-bar = -partial-bar partial. The relation partial-bar^2 = 0 is what makes partial-bar into a differential and lets us form Dolbeault cohomology. Crucially, all of this rests on integrability (Newlander-Nirenberg); on a non-integrable almost-complex manifold partial-bar^2 is generally nonzero.
Why they matter: holomorphic functions and sections are precisely the ones killed by partial-bar, so solving the inhomogeneous equation partial-bar u = f (the 'del-bar problem') is the central analytic engine of the subject — it produces holomorphic functions with prescribed behaviour, underlies the holomorphic Poincaré (Dolbeault-Grothendieck) lemma, and via L^2 estimates (Hormander, Kohn) drives existence theorems for sections of line bundles. A common confusion: partial-bar f = 0 characterizes holomorphic f, but partial-bar applied to a higher (p,q)-form measures something cohomological, not 'holomorphy', and partial is not the complex conjugate of partial-bar unless one is careful about which bundle one acts on.
On C with coordinate z, take f(z, z-bar) = z (z-bar) = |z|^2. Then partial f = (z-bar) dz and partial-bar f = z d(z-bar). Since partial-bar f is not zero, |z|^2 is not holomorphic — correct, it is real-valued and nonconstant. By contrast f(z) = z^2 has partial-bar f = 0, confirming it is holomorphic.
partial-bar f = 0 is the Cauchy-Riemann test: |z|^2 fails it, z^2 passes it.
partial-bar^2 = 0 (needed to define cohomology) holds because the complex structure is integrable. On a merely almost-complex manifold partial-bar^2 need not vanish, so there is no Dolbeault cohomology in the usual sense.