Let $\pi:\mathcal X\to B$ be a family of Kaehler manifolds then if we take $B'\subset B$ be the set of parameters such that $X\_b$ admit Kaehler-Einstein metric(with zero, negative, or positive Ricci curvature) , then $B'$ is Zariski open subset of $B$ always?



Zariski open subset on family of Kaehler-Einstein manifolds
The answer depends on diameter bounds of the fibers $(X\_b,\omega\_b)$ and the $L^\infty$ norm of the Ricci curvature. (You can derive the answer by using a nice paper of J. Cheeger-A.Naber. See the nice survay paper of [Donaldson](http://www.intlpress.com/site/pub/pages/journals/items/sdg/content/vols/0019/0001/a005/).) When $b\to 0$, the diameter of $X\_b$ may blow up near the central fiber. If such an assumption is satisfied then $B'$ is a Zariski open subset of $B$. 


When fibers $X\_b$ are Calabi-Yau manifolds, then, fixing holomorphic $n$-forms $\Omega\_b$ on $X\_b$, $b\neq 0$, which vary holomorphically in $b$, we may define the Weil-Petersson pseudometric on $Δ^×$ by [Tian's formula](https://hal.archives-ouvertes.fr/hal-01413746v4/document):


$$\omega\_{WP}=-\sqrt{-1}\partial\_b\bar\partial\_b\log\left((-1)^{n^2/2}\int\_{X\_b}\Omega\_b\wedge\overline {\Omega\_b}\right),$$


which is a smooth semipositive definite Hermitian form on $Δ^×$, well-defined independent of the choice of $\Omega\_b$. (It would be nice to find the same formula when dimension of base is bigger than one.)


Then boundedness of diameter of $X\_b$ corresponds to boundedness of $$\int\_{X\_b}\Omega\_b\wedge\overline {\Omega\_b}<C$$ which is equivalent to the central fiber having canonical singularites. See [Takayama's](http://www.ms.u-tokyo.ac.jp/journal/abstract_e/jms220115_e.html) paper.


When the dimension of base is bigger than one, then instead of boundedness of diameters you need to take the boundedness of the Song-Tian-Tsuji measure $$\omega\_{SRF}^{n-m}\wedge \pi^\*\omega\_B^m<C.$$


Here $\omega\_{SRF}$ is the semi flat metric of Vafa-Yau et al. and I assume that your base admits a Kahler Einstein metric $Ric(\omega\_B)=\lambda\omega\_B$.


You can find more information in the papers of [Odaka](https://arxiv.org/abs/1211.4833) , Tian, and Donaldson.


In general boundedness of diameter of fibers corresponds to Weil-Petersson distance $d\_{WP}(B,0)<C$. In fact in the algebraic geometric view, it corresponds to birational modification and semi-stable reduction (due to Wang and Tosatti).


You need to take the group of holomorphic automorphisms of each $X\_b$ to be discrete. (See the example of Tian about Mukai-Umemura manifolds.)


