---
tag: Y3900T2EY5
kind: concept
name: Dolbeault cohomology
msc: [32L10, 32C35, 58A10]
sources:
  - url: https://en.wikipedia.org/wiki/Dolbeault_cohomology
same_as:
  wikidata: Q1235317
  wikipedia: Dolbeault cohomology
---

## Summary

Dolbeault cohomology is the cohomology of the complex of smooth [[(p,q)-forms|DC81TCR98Y]] on a [[complex manifold|JSGWAN2RJV]] with the [[dbar operator|M1VKCW3RFW]] as differential.
It measures to what extent the equation $\bar\partial u = f$ fails to be solvable for $\bar\partial$-closed data $f$.
By the [[Dolbeault isomorphism|G2HC1SFQ4P]] it computes the sheaf cohomology of the sheaves of holomorphic $p$-forms.
On compact Kähler manifolds it gives the Hodge decomposition of de Rham cohomology.

## Explanations

### L1 human

On a complex manifold, a differential form splits into pieces of type $(p,q)$: $p$ holomorphic differentials $dz_j$ and $q$ antiholomorphic differentials $d\bar z_k$.
The operator $\bar\partial$ differentiates in the antiholomorphic directions and squares to zero.
Dolbeault cohomology is the space of $\bar\partial$-closed forms modulo $\bar\partial$-exact forms.
For example, $H^{0,0}$ is the space of holomorphic functions, since a function with $\bar\partial f = 0$ is holomorphic.
On a polydisc in $\C^n$ all groups $H^{p,q}$ with $q \ge 1$ vanish, while on a compact Riemann surface of genus $g$ the group $H^{0,1}$ has dimension $g$.