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Let $T$ be a compact torus, and $X$ its blow-up in a point (or in several points). It seems that $X$ is K-stable for any Kahler form on $X$. Is there a reference to this?

Also, what can we say about the constant scalar curvature Kahler metrics on this blow-up? Do they exist in all Kahler classes? They do exist in some Kahler classes, by the famous result of Arezzo and Pacard:

Claudio Arezzo and Frank Pacard, Blowing up and desingularizing constant scalar curvature Kähler manifolds, Acta Math. Volume 196, Number 2 (2006), 179-228.

If I am right and the standard conjectures about K-stability are true, it seems that the CSC Kahler metrics should exist for all Kahler classes.

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This statement about blow ups of torus is not correct. Take any aglebraic $2$-torus with a smooth curve $C$ of genus $>1$. Blow up $C^2+1$ points on $C$ and apply Theorem 1 here: https://arxiv.org/pdf/0710.4078.pdf

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  • $\begingroup$ and what if the torus is non-algebraic? $\endgroup$
    – Misha Verbitsky
    Commented Jun 29, 2017 at 5:00
  • $\begingroup$ For the application I have in mind, non-algebraic tori are more than sufficient $\endgroup$
    – Misha Verbitsky
    Commented Jun 29, 2017 at 5:02
  • $\begingroup$ I would bet that no positive result in this direction that is stronger than Arezzo-Pacard is known. I would guess (but this is only a guess) that using Theorem 1.2 here arxiv.org/pdf/0807.1716.pdf one could construct example similar to what I gave you by making an iterated blow up of a non-algebraic torus. This would lead to appearance of quite "bad" divisors there, composed of rational curves. But probably you should ask experts like Yuji Odaka, or Gabor Szekelyhidi $\endgroup$
    – aglearner
    Commented Jun 29, 2017 at 10:21

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