If we are given compact complex manifold $X$ and a Kahler class $[\omega]$, 
can we always find a positive definite representative $\omega \in [\omega]$ that is 
real analytic?



Analytic representatives for Kahler classes
Yes. Run the Kähler-Ricci heat flow 
$$
\frac{\mathrm{d}}{\mathrm{d}t}\left(\omega(t)\right) = -\mathrm{Ric}\bigl(\omega(t)\bigr)
$$ 
with initial condition $\omega(0) = \omega$. This will exist for some time interval $[0,T)$, and the $\omega\_t$ for $t>0$ will all be be real-analytic with respect to the natural real-analytic structure on $M$ given by the complex structure. Note that we have
$$
\left[\omega(t)\right] = \left[\omega(0)\right] - t\,c\_1(M).
$$


Now, for $0<t\_1<t\_2<T$, with $t\_1$ very small with respect to $t\_2$, consider the $2$-form
$$
\omega(t\_1,t\_2) = \frac{\omega(t\_1) - (t\_1/t\_2)\ \omega(t\_2)}{1-t\_1/t\_2}
$$
It is easy to see that, if $t\_1/t\_2>0$ is very small, then $\omega(t\_1,t\_2)$ is a positive $(1,1)$-form. It is real-analytic and, by construction, it satisfies
$$
\left[\omega(t\_1,t\_2)\right] = \left[\omega(0)\right] = [\omega].
$$


