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The question is stated in the title. Could one introduce all the complex geometry concepts just by using the "anti-" objects instead? Or is it just a $Z_2$ symmetry thing like it sometimes is with complex problems?

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    1) The identity map is holomorphic; it is not antiholomorphic. 2) If you pull a $(p,q)$-form by a holomorphic map, you get another $(p,q)$-form; but pulling it by an antiholomorphic map gives you a $(q,p)$ form, an element of a different space. Etc. It's like comparing positive numbers to negative: as long as you stick to addition, there is a perfect symmetry, but once you start considering multiplication, positive numbers win.2012-06-07

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  1. The identity map is holomorphic; it is not antiholomorphic.

  2. If you pull a (p,q)-form by a holomorphic map, you get another (p,q)-form; but pulling it by an antiholomorphic map gives you a (q,p) form, an element of a different space. Etc. It's like comparing positive numbers to negative: as long as you stick to addition, there is a perfect symmetry, but once you start considering multiplication, positive numbers win.

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    Very nice and crisp explanation.2012-08-07