Interpretiing the Dolbeault operators

[lavinia]

The differential of a function may me interpreted a the the dual of its gradient.

What is the interpretation of the Dolbeault operators?

[zhentil]

Hmm, I'd prefer to interpret the differential of a function as a map on tangent spaces. Then the Dolbeault operators further clarify the interaction between the map and complex multiplication. Recall that a map from C to C is holomorphic if and only if \frac{d}{d\bar{z}}f(z)=0. Phrased another way, the Jacobian of the map commutes with multiplication by i. If we then look at a map from C to C which is not necessarily holomorphic, we can decompose its Jacobian into components which commute/anticommute with complex multiplication.

Of course, one may attempt to extend this to your case, by defining holomorphic gradients and the like, but IMO it's clearer this way.

[lavinia]

Hmm, I'd prefer to interpret the differential of a function as a map on tangent spaces. Then the Dolbeault operators further clarify the interaction between the map and complex multiplication. Recall that a map from C to C is holomorphic if and only if \frac{d}{d\bar{z}}f(z)=0. Phrased another way, the Jacobian of the map commutes with multiplication by i. If we then look at a map from C to C which is not necessarily holomorphic, we can decompose its Jacobian into components which commute/anticommute with complex multiplication.

Of course, one may attempt to extend this to your case, by defining holomorphic gradients and the like, but IMO it's clearer this way.

Thanks. I got the thought - maybe wrong - that in a particular conformal atlas of charts on a Riemann surface - the operation adx + bdy -> -bdx + a dy is well defined. It is something like a rotation by 90 degrees but there is no Riemannain metric. I guess it would be a rotation in isothermal coordinates.