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I have a question related to the preprint "Heights and metrics with logarithmic singularities" by G. Freixas i Montplet.

Let $X$ be an arithmetic variety with arithmetic divisor $D$ how can we introduce an arithmetic divisor with conic singularities? Here is the definition of conical singularities on a Kähler variety

I am looking for an arithmetic version.

If an arithmetic variety be of general type can we say $Ric(\omega)=-\omega$ on $X(\mathbb C)$ ? What about the definition of Kähler-Einstein metric on arithmetic varieties? When does it exist?

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  • $\begingroup$ I wonder, what is the interest in conic singularities in the arithmetic world? They are useful for continuity methods in Kaehler geometry. $\endgroup$
    – Ruadhaí Dervan
    Commented Feb 16, 2016 at 12:01
  • $\begingroup$ see paper of Odaka about arithmetic canonical metric $\endgroup$
    – user21574
    Commented Feb 16, 2016 at 12:33
  • $\begingroup$ He just mentions that one could extend the theory, he does not motivate why one would be interested. $\endgroup$
    – Ruadhaí Dervan
    Commented Feb 16, 2016 at 12:53
  • $\begingroup$ Actually I am readding the paper of Odaka and I faced with this question $\endgroup$
    – user21574
    Commented Feb 16, 2016 at 13:42

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