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Why do so many textbooks define holomorphic functions $f: \mathbb{C}^n \to \mathbb{C}$ as $\mathbb{R}$-differentiable functions satisfying $$\frac{\partial f}{\partial \overline{z_i}}=0 $$ where we use Wirtinger operators

$$\frac{\partial}{\partial z_i} := \frac{1}{2} (\frac{\partial}{\partial x_i} - i \frac{\partial}{\partial y_i}) \ \ \text{and} \ \ \frac{\partial}{\partial \overline{z_i}} := \frac{1}{2} (\frac{\partial}{\partial x_i} + i \frac{\partial}{\partial y_i}) \ ?$$

To me defining $\frac{\partial}{\partial z_i}$ in this manner looks very artificial.

A much better approach seems to be using the usual definition of differentiability of maps between affine spaces over $\mathbb{C}$, that is requiring $$f(z-z_0)=L_{z_0} \cdot (z-z_0) + o( \| z - z_0 \|)$$hold for all $z_0$ in the domain we are interested in and where $L_{z_0}: \mathbb{C}^n \to \mathbb{C}$ is complex linear. Then we have the usual Jacobi matrix where $\frac{\partial f}{\partial z_i}$ has its usual meaning as $$\frac{\partial f}{\partial z_i} |_{z_0} = \lim_{h \to 0} \frac{f(z_0 + h\cdot \vec{e_i}) - f(z_0)}{h}, \ \ h \in \mathbb{C}.$$

Then we of course observe that if $f$ is $\mathbb{C}$-differentiable then it is also $\mathbb{R}$-differentiable and we just have the equality: $$\frac{\partial f}{\partial z_i} = \frac{1}{2} (\frac{\partial f}{\partial x_i} - i \frac{\partial f}{\partial y_i}).$$

This way both sides of the equation have their natural meaning and we are not defining the LHS via the RHS. An advantage of this approach is that we no longer have to perform ad hoc checks of chain rules for Wirtinger operators and the composition of holomorphic functions being again holomorphic becomes a triviality.

As another advantage it seems that if one is not interested in Hodge theory and one just wants to define holomorphic tangent bundle and holomorphic bundle of one-forms for a complex manifold there is no longer need to go through the yoga of defining $(1,0)$ and $(0,1)$ splittings and one can proceed in the usual differential-geometric way, defining the holomorphic tangent sheaf as the sheaf of $\mathbb{C}$ derivations of the structure sheaf with the local holonomic basis consisting of $\{ \frac{\partial}{\partial z_i} \}$.

To sum it up, is that just a historical way to deal with holomorphic functions as real ones and use Wirtinger operators or is there some important point I am missing here?

One possible explanation I can think of is that for a complex manifold those $(p,q)$ decompositions (or more generally the interplay between real and holomorphic) are so much more interesting that people care much less about proper holomorphic objects and hence don't bother laying out a streamlined analytic exposition of them.

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  • $\begingroup$ I don't think I've ever seen a text that defines holomorphicity that way. $\endgroup$ – David C. Ullrich Nov 19 '17 at 16:22
  • $\begingroup$ Well yes, maybe they define it using convergent power series, but my question is why there is so much emphasis put on Wirtinger symbols $\endgroup$ – Bananeen Nov 19 '17 at 19:37
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Many textbooks define holomorphy of a function $f: \mathbb{C}^n \to \mathbb{C}$ to be a function that may be written as a sum of multiple power series $$f(z) = \sum_{\nu_1 \cdots \nu_n =0}^{\infty} c_{\nu_1 \cdots \nu_n} (z_1 - w_1)^{\nu_1} \cdots (z_n - w_n)^{\nu_n}.$$

This seems most clear to me.

As you have written above, Wirtinger operators are convenient for decomposing the bundles. Each have their merits, going back to the standard limit definition however seems to be a hassle considering the machinery that is developed in a standard first course.

Which reference are you looking at? My favourite is Shabat's Introduction to Complex Analysis - Part II.

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  • $\begingroup$ Shabat's seems to be one of the few textbooks which mentions the usual analytic definition. The ones I have in mind are Huybrecht's "Complex geometry" and Taylor's "Serveral complex variables with Connections to Algebraic Geometry and Lie Groups". Both discuss the Cauchy-Riemann conditions (via those Wirtinger symbols) and power series, but don't emphasize the standard analytic one $\endgroup$ – Bananeen Nov 19 '17 at 19:36
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    $\begingroup$ Hmmm I am not a huge fan of Huybrecht's Complex Geometry. Personally, I feel that he mentions many results without properly emphasising the motivation/difficulty behind the ideas. Other good books include Gunning and Rossi's Analytic Functions of Several Complex Variables and From Holomorphic Functions to Complex Manifolds, I forget the authors. $\endgroup$ – AmorFati Nov 20 '17 at 0:44
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    $\begingroup$ I guess it depends on what the authors are attempting to do. For example, Huybrecht's is doing complex geometry, so he wants to get to tangent spaces and bundles as soon as possible. Shabat however is interested in the complex analysis itself, and therefore, I guess, takes his time, laying all the ground work. Does this answer your question? $\endgroup$ – AmorFati Nov 20 '17 at 0:57

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