Dolbeault operator

Summary

The Dolbeault operator $\bar\partial$ on a complex manifold is a part of the exterior derivative. It is the part that raises the antiholomorphic degree of a complex differential form by one. A function is holomorphic 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, and its index is the holomorphic Euler characteristic given by the Riemann–Roch theorems.