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Let $M$ be a complex manifold. Then there are three different notions of a tangent space on $M$;

  1. The real tangent space locally generated (on some coordinate patch) over smooth real functions locally on a coordinate patch by $\partial_{x}, \partial_y$

  2. The complexified tangent space locally generated over smooth complex functions by $\partial_z, \partial_{\overline{z}}$

  3. The holomorphic tangent space locally generated over holomorphic function by $\partial_z$

Then, dually, there exist three types of cotangent spaces on a complex manifold.

Roughly, in the smooth case, if $M$ has dimension $n$, then we integrate over $M$ by integrating (global) sections of $\Lambda^n T_M^*$.

In the complex case, what is the correct notion of integration?

It seems like we have a few options here.

Gibbs
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Yuugi
  • 2,143
  • Remember that an $n$-dimensional complex manifold will be a $2n$-dimensional real manifold, and so you'll want to integrate $2n$-forms. In the complexified setting, these will be forms of type $(n,n)$ (i.e., function multiples of $(\frac i2)^ndz^1\wedge d\bar z^1\wedge\dots\wedge dz^n\wedge d\bar z^n$). – Ted Shifrin Apr 17 '18 at 22:26
  • @TedShifrin Hi I was hoping you might be able to help me with a question I had, if you have any time. Thanks! – bowlofpetunias May 22 '20 at 16:57

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