The Dolbeault operators on a complex manifold are the components 
and 
of the exterior derivative after differential
forms are decomposed by type, so 
. They satisfy 
and 
.
Dolbeault Operators
See also
Del Bar Operator, Differential k-Form, Dolbeault Cohomology, Exterior DerivativeExplore with Wolfram|Alpha
References
Wells, R. O. Jr. Differential Analysis on Complex Manifolds, 3rd ed. New York: Springer-Verlag, 2008.Cite this as:
Weisstein, Eric W. "Dolbeault Operators." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/DolbeaultOperators.html