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?
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1$\begingroup$ Are you assuming $B$ is $\mathbb{C}^n$ for some n? Otherwise, what do you mean by Zariski open? In general Kähler manifolds are not algebraic, so there is no meaningful way to talk about a Zariski open set. Additionally it seems you have stated your assumption about the fibers incorrectly, because with what you have written currently, you are already assuming all the fibers admit Kähler-Einstein metrics (so $B'=B$). $\endgroup$– A. S.Commented Jul 26, 2017 at 1:03
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3$\begingroup$ @A.S.: I think he means open in the complex analytic Zariski topology: the topology whose closed sets are the closed analytic subvarieties. $\endgroup$– Ben McKayCommented Jul 26, 2017 at 6:42
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1 Answer
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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.) 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:
$$\omega_{WP}=-\sqrt{-1}\partial_b\bar\partial_b\log\left((-1)^{n^2/2}\int_{X_b}\Omega_b\wedge\overline {\Omega_b}\right),$$
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 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 , 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.)
