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Let $\mathcal X\to \mathcal S$, be a family of polarized Kaehler manifolds with $\omega_s= Ric(\omega_s)$(i.e., fibers are Fano Kahler-Einstein manifolds). Then $dim Aut(X_s)=Const$.?

Is there any construction of the coarse moduli space of compact polarized Fano Kaehler-Einstein manifolds?

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  • $\begingroup$ Consider a Fano manifold $X$ such that ${\rm Aut}(X)$ is transitive on an open set $U\subset X$ and fixes a point $x_0\in X$. If you blow-up $x_0$ then all the automorphisms will lift to the blow-up, if you blow-up a point of $U$ youre potentially decreasing the dimension of the automorphism group. Take the family of $X$ blown-up at $p$ where $p$ moves from $U$ to $x_0$ and then find a new polarization in this family. $\endgroup$
    – Joaquín Moraga
    Sep 30, 2017 at 18:44
  • $\begingroup$ It is not true that the automorphism group stays constant in flat families. For an explicit example, look at the degenerations of Fano threefolds with a two dimensional torus action to toric Fano threefolds (the papers of Ilten-Süss). For the construction of the moduli space, see the work of Odaka and Li-Wang-Xu. $\endgroup$
    – Ruadhaí Dervan
    Sep 30, 2017 at 23:37
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    $\begingroup$ In your case $s\to \text{ord Aut}(X_s)$ is upper semicontinuous. Let $Aut^ω(\mathcal X/S)$ denote the space of automorphisms of $X_s$ which fix the Kähler classes $[ω_{X_s}]$ then if fibers admit Kähler -Einstein metric $Ric(\omega_s)=\omega_s$, then $\text{Aut}^ω(X/S)\to S $ is proper . The proof is as same as the paper of link.springer.com/article/10.1007/BF01389100 $\endgroup$
    – user21574
    Dec 4, 2017 at 3:36
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    $\begingroup$ Let $X→S$ and $Y→S$ be families of compact complex manifolds whose fibers admit Kähler -Einstein metric with positive Ricci curvature; for every $s∈S$ let $X_s$ and $Y_s$ be isomorphic. Then $X→S$ and $Y→S$ are locally isomorphic. You can adopt the same proof of link.springer.com/article/10.1007/BF01389100 . see link.springer.com/article/10.1007%2FBF01360031?LI=true and link.springer.com/article/10.1007%2FBF01589190?LI=true $\endgroup$
    – user21574
    Dec 4, 2017 at 3:46
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    $\begingroup$ Note that my previous comments are about the order of Automorphism group of fibers and is not directly related to your question, but it may be your interest. $\endgroup$
    – user21574
    Dec 4, 2017 at 4:48

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