If $\omega \in F^r$ is complex r-form then we can define complex conjugate element $\bar\omega \in F^r$ by setting

$\bar\omega(v_1,.....v_r)= \overline{\omega(v_1,......,v_r)}$ where $v_1,\ldots,v_r$ are coming from tangent space.

So if $\omega$ is $(p,q)$ form then $\bar\omega$ is $(q,p)$ form.

Now $dz$ is $(1,0)$ form so $d\bar z= \overline {dz}$ is $(0,1)$ form.

Hence $d\bar z(\frac {\partial}{\partial{\bar z}})=\overline {dz(\frac {\partial}{\partial{\bar z}})}=\bar 0=0$.

But it should be $1$ as $d\bar z$ is element of dual basis w.r.t. basis $(\frac {\partial}{\partial{z}},\frac {\partial}{\partial{\bar z}})$.

Then what mistake I am making?

  • $\begingroup$ @totoro I am following 'From holomorphic functions to complex manifolds' by Fritzsche and Grauert in which it says $d\bar z= \overline {dz}$ $\endgroup$
    – Mayuresh L
    May 2 '18 at 15:42

In Fritzsche & Grauert's definition of the complex conjugate of a form, they are assuming $v_1,\dots,v_r$ are elements of the tangent space, meaning they are real vectors. If you allow them to be complexified tangent vectors, then you have to write $$\bar\omega(v_1,.....v_r)= \overline{\omega(\bar v_1,......,\bar v_r)}.$$ If you insert this into your computation, you'll see that it fixes the problem.

Added: Fritzsche & Grauert define a complex $r$-form at $x\in M$ as an alternating multilinear map $$ \omega\colon T_xM \times \dots \times T_xM \to \mathbb C. $$ Any such form can be decomposed into its real and imaginary parts: $\omega = \alpha + i\beta$, where $\alpha$ and $\beta$ are ordinary real-valued $r$-forms. If you plug this into the definition of $\bar\omega $, you'll find that $\bar \omega = \alpha - i\beta.$

If $z = u+iv$ is a complex-valued smooth function, then $dz$ is by definition the complex $1$-form defined by $dz = du + i \, dv$. Thus $d\bar z = d (u-iv) = du - i\,dv = \overline{dz}$.

Now if $z^j=x^j+iv^j$ is one of the complex coordinate functions in a holomorphic coordinate chart $(z^1,\dots,z^n)$, then the complex vector fields $\partial/\partial z^j$ and $\partial/\partial \bar z^j$ are defined by $$ \frac{\partial}{\partial z^j} = \frac{1}{2} \left( \frac{\partial}{\partial x^j} - i \frac{\partial} {\partial y^j}\right), \qquad \frac{\partial}{\partial \bar z^j} = \frac{1}{2} \left( \frac{\partial}{\partial x^j} + i \frac{\partial} {\partial y^j}\right). $$ Then $\overline{\partial /\partial z^j} = \partial /\partial \bar z^j$, and you can check that the definitions yield $$ dz^j \left( \frac{\partial}{\partial z^j} \right) =1 ,\qquad dz^j \left( \frac{\partial}{\partial \bar z^j} \right) =0, $$ along with the analogous formulas for $d\bar z^j$ obtained by conjugation: $$ d\bar z^j \left( \frac{\partial}{\partial \bar z^j} \right) =1 ,\qquad d\bar z^j \left( \frac{\partial}{\partial z^j} \right) =0. $$

  • $\begingroup$ Ohh...thanks a lot! So in general is it true that $d\bar z= \overline {dz}$? What is definition of $d\bar z$ in general? Please elaborate little more. This is confusing me since last two weeks. Thanks again!! $\endgroup$
    – Mayuresh L
    May 2 '18 at 19:34
  • $\begingroup$ @MayureshL: See my addition above. $\endgroup$
    – Jack Lee
    May 2 '18 at 20:16
  • $\begingroup$ But my question was how things changes in case of $d\bar z$ when we have complexified tangent space. Anyway Thanks $\endgroup$
    – Mayuresh L
    May 2 '18 at 20:21
  • $\begingroup$ To summarise my question is....What are anallogous formulas for $d\bar z^j$ in Fritzsche and Grauert definition and when we consider complexified tangent bundle? $\endgroup$
    – Mayuresh L
    May 2 '18 at 20:31
  • $\begingroup$ @MayureshL: See the last line added above. Does that answer your question? $\endgroup$
    – Jack Lee
    May 2 '18 at 20:56

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