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Is it currently known whether or not any the standard ball quotient models (As introduced in Allcock-Carlson-Toledo, Laza, Yokoyoma,... is an example of a moduli space of K-polystable Fano varieties (or equivalently - arising from the theory of Fano varieties equipped with a Kähler-Einstein metric)?

In fact - when $dim=2$, there is the Odaka-Spotti-Sun moduli of Del-Pezzo surfaces. What is an example of a moduli space of K-polystable Fano varieties of dimension >2 that we understand explicitly?

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  • $\begingroup$ Do we know of any Fano hypersurface of degree $d\geq 3$ that is proved to be $K$-stable? $\endgroup$
    – Jason Starr
    Commented Mar 10, 2017 at 16:02
  • $\begingroup$ I don't know ... that's why I'm asking :-) , this moduli spaces were conjectures to be such examples a few years ago ... I was wondering if that conjecture was proven. $\endgroup$
    – Nati
    Commented Mar 10, 2017 at 16:36
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    $\begingroup$ For cubic 3-fold, see my paper with C. Xu: projecteuclid.org/euclid.dmj/1562033045 $\endgroup$
    – Yuchen Liu
    Commented Sep 28, 2019 at 3:42

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