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page 6 in this notesays:

A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components.

I never knew there were holomorphic components, I only was aware of holomorphic mappings and holomorphic functions. Can someone please point out why was this terminology used here?

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PhilosophicalPhysics
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    This is one of many abuses of terminology in play in complex geometry. Consider $\mathbb{C}^n$. It's a complex vector space of dimension $n$, so it has a basis $(z_1, \ldots, z_n)$. It's also a real vector space of (real) dimension $2n$, but we need $n$ more vector to get a real basis. Those are $(z_1, \ldots, z_n, \bar z_1, \ldots, \bar z_n)$ (it's maybe clearer to write $z_j = x_j + i y_j$ etc to see this, where the $x_j, y_j$ are a real basis of $\mathbb{R}^{2n}$). People now often speak of the $z_j$ as being "holomorphic" and the $\bar z_j$ as "antiholomorphic". – Gunnar Þór Magnússon Dec 18 '15 at 17:03
  • You should be able to make sense of the quote you posted by translating this to the tangent space of a manifold. – Gunnar Þór Magnússon Dec 18 '15 at 17:04
  • @GunnarÞórMagnússon Aha, so in complex manifolds, it is not really common to use "holomorphic" as a description of $z_j$, nor is it common to describe $\bar{z}_j$ as "antiholomorphic", but people just go ahead and use them. This confuses me, though, because I cannot make sense out of it. That is, why are they being called like that? I know a holomorphic function is an analytic function, but how would I understand "holomorphic" and "antiholomorphic" coordinates? – PhilosophicalPhysics Dec 18 '15 at 18:07
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    Well, you can sort of motivate it like this: take a random power series and write it in powers of $z_j$ and $\bar z_k$ instead of powers of $x_j, y_k$. Then the function that power series defines a holomorphic function if and only if you can write it only with the "holomorphic" z_j coordinates. (Alternatively, the coordinate functions $z_j$ are holomorphic and the $\bar z_k$ are antiholomorphic.) – Gunnar Þór Magnússon Dec 18 '15 at 18:46
  • Thanks! @GunnarÞórMagnússon – PhilosophicalPhysics Dec 19 '15 at 18:24
  • Related: (Anti-) Holomorphic significance? – Andrew D. Hwang Jan 03 '16 at 00:57

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