Dbar operator

Summary

The $\bar\partial$ operator, read "d-bar", sends a $(p, q)$-form to a $(p, q+1)$-form by differentiating its coefficients in the conjugate variables $\bar z_1, \dots, \bar z_n$. On functions it is $\bar\partial f = \sum_j \frac{\partial f}{\partial \bar z_j} \, d\bar z_j$, and $\bar\partial f = 0$ is exactly the system of Cauchy–Riemann equations in several variables. It satisfies $\bar\partial \circ \bar\partial = 0$ and splits the exterior derivative as $d = \partial + \bar\partial$. It generalises the one-variable Cauchy–Riemann operator, and it is the basic operator of Dolbeault cohomology and of the $L^2$ methods of several complex variables.