I am trying to read Kobayashi's "Differential geometry of complex vector bundles". There are many places where a complex differential form is referred to as being real. e.g
Chapter I, Proposition 7.24 p. 28 " A closed real $(p,p)$ form $\omega$ on a compact Kähler manifold M is cohomologous to zero if and only if $ \omega = id' d'' \phi $ for some real $(p-1,p-1)$-form $\phi. $ "
Similarly, p. 41 Chapter II, Proposition 2.23 " Given any closed real $(1,1)$-form $\phi$ representing $c_{1}(E),$ there is an Hermitian structure $h$ in $E$ such that $\phi = c_{1}(E)$ provided $M$ is compact Kähler."
What does it mean? Does it mean that the coefficients are real valued function? Or does it mean that it is the same under complex conjugation? i.e. $\bar{\phi} = \phi ? $
To be more explicit,
Let $\alpha, \beta : \mathbb{C}^{2} \rightarrow \mathbb{R}$ be real valued smooth functions and let $$ \phi = [\alpha(z_{1}, z_{2}) + i \beta(z_{1}, z_{2})] dz_{1}\wedge d\bar{z_{2}} - [\alpha(z_{1}, z_{2}) - i \beta(z_{1}, z_{2})] dz_{2}\wedge d\bar{z_{1}} . $$
This is a $(1,1)$-form on $\mathbb{C}^{2}$ with the property that $\bar{\phi} = \phi, $ since $ \overline{dz_{1}\wedge d\bar{z_{2}}} = - dz_{2}\wedge d\bar{z_{1}},$ but this is a real valued form only if $\beta \equiv 0$ on $\mathbb{C}^{2}.$ Or does $\phi$ being real mean something else?
Thanks in advance for any help.
That means if you write $dz_j=dx_j+\sqrt{-1}dy_j$ and expand everything you are left with only real coefficients. For example, a $(1,1)$-form $\xi dz\wedge d \overline{z}$ is real if writing $\xi=\alpha+\sqrt{-1}\beta$ and $dz=dx+\sqrt{-1}dy$,$$\xi dz\wedge d\overline{z}=(\alpha+\sqrt{-1}\beta)(-2\sqrt{-1}dx\wedge dy)$$ has real coefficient, i.e., $(\alpha+\sqrt{-1}\beta)(-2\sqrt{-1})$ is a real function.