---
tag: M2F4N31JQ2
kind: concept
name: Dolbeault operator
msc: [32W05, 58J10, 32Q99]
sources:
  - url: https://en.wikipedia.org/wiki/Complex_differential_form
---

## Summary

The Dolbeault operator $\bar\partial$ on a [[complex manifold|JSGWAN2RJV]] is a part of the [[exterior derivative|0ZHSTAG8ZY]].
It is the part that raises the antiholomorphic degree of a complex [[differential form|0E3G0GRPZ6]] by one.
A function is [[holomorphic|0NAHXR1FRH]] exactly when $\bar\partial f = 0$, so $\bar\partial$ is the Cauchy–Riemann operator of complex geometry.
It satisfies $\bar\partial^2 = 0$, and the cohomology of the resulting Dolbeault complex computes the sheaf cohomology of holomorphic forms.
The Dolbeault complex is an [[elliptic complex|M8GMH8GQR6]], and its index is the holomorphic Euler characteristic given by the Riemann–Roch theorems.

## Explanations

### L1 human

In one complex variable $z = x + iy$, a function is holomorphic when it satisfies the Cauchy–Riemann equations.
These can be written as one equation $\partial f / \partial \bar z = 0$, where $\partial / \partial \bar z = \frac{1}{2}(\partial_x + i \partial_y)$.
The Dolbeault operator packages this derivative as $\bar\partial f = \frac{\partial f}{\partial \bar z} \, d\bar z$, and extends it to forms and to several variables.
For example $\bar\partial z = 0$, since $z$ is holomorphic, while $\bar\partial \bar z = d\bar z \ne 0$.