dbar2

dbar2 is not a standard, widely recognized term on its own. In many mathematical contexts, it can be read as the square of the Dolbeault operator, denoted ∂̄ or d̄. The expression d̄^2 (or ∂̄^2) represents applying the Dolbeault operator twice. On any complex manifold with an integrable complex structure, this composition vanishes identically: (∂̄)^2 = 0. Because of this nilpotency, d̄^2 does not define a new differential operator, but rather expresses a fundamental property of the Dolbeault framework.

In complex geometry, the Dolbeault operator ∂̄ acts on smooth forms of type (p,q), mapping them to (p,q+1)-forms. The condition ∂̄^2 = 0 yields a differential in the ∂̄-complex, and its cohomology groups H^{p,q}_{∂̄}—the Dolbeault cohomology groups—measure the failure of ∂̄-exactness and encode complex-analytic information about the manifold. These groups play a central role in several complex variables, complex manifolds, and Hodge theory, and they interact with the de Rham cohomology through the overall d = ∂ + ∂̄ decomposition of the exterior derivative.

If you encounter the term dbar2 outside this mathematical context, it may be a shorthand or project-specific name rather than a standard mathematical object. In absence of context, the safest interpretation is that it refers to the square of the d̄ (Dolbeault) operator, which is zero, rather than to a distinct operator. See also Dolbeault operator, Dolbeault cohomology, and complex manifold.