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Is there any approach for the following conjecture?

Let $X$ be a projective Fano manifold and let $Pic(X) \cong \mathbb Z$, then $(X, −K_X) $ is $K$-semistable.

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    $\begingroup$ K-stability, as any other notion of stability, applies to polarized varieties. Are you assuming that $-K_X$ is ample? $\endgroup$ – abx Nov 5 '15 at 10:11
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    $\begingroup$ Yes, $-K_X$ is ample here, X is Fano $\endgroup$ – user21574 Nov 5 '15 at 10:18
  • $\begingroup$ That conjecture is false. There is the Pasquier-Perrin example that abx first mentioned here some time ago. $\endgroup$ – Jason Starr Nov 5 '15 at 10:46
  • $\begingroup$ The article I refer to is as follows, "Local rigidity of quasi-regular varieties" by Pasquier and Perrin, Math. Z. 265 (2010), no. 3, 589–600. Now I need to double-check that the example is not K-semistable; I am certain that it is not K-polystable, e.g., by Li-Wang-Xu. I will double-check with my colleagues about K-semistability. $\endgroup$ – Jason Starr Nov 5 '15 at 11:22
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    $\begingroup$ My colleague says that the parahoric limit in the Pasquier-Perrin example cannot be $K$-semistable. $\endgroup$ – Jason Starr Nov 5 '15 at 15:19
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The answer correspond to weighted stability of Fedor Bogomolov and result of Gang Tian. We explain it here as an additional comment to previous answer

Tian introduced a stronger notion of strong stability around 90 as follows and proved the following theorem albeit strong stability does not satisfied for any Fano manifold. But it can has its own interest.

Definition(Tian 92): Let $E_1$ and $E_2$ be two coherent holomorphic sheaves on $X$. An extension of $E_1$ by $E_2$ is a coherent sheaf $E_3$ with the following short exact sequence $$0\to E_2\to E_3\to E_1\to 0$$

A pair $(E_1,E_2)$ of coherent sheaves is said to be stable (resp semi-stable) with respect to Kahler class $\omega$ if the generic extension $R$ of $E_1$ by $E_2$ is stable(resp semi-stable) with respect to same Kahler class.

Now let $E$ be a holomorphic vector bundle then we say $E$ is strongly stable (resp semi-stable)with respect to $\omega$ if both $E$ and the pair $(E,\mathcal O_X)$ are stable with respect to $\omega$. Here $\mathcal O_X$ is the structure sheaf of $X$. i.e sheaf of local holomorphic functions.

Now Bogomolov introduced the following $\mu$-stability as weighted stability.

Definition: Given any positive number $\mu<1$. A bundle $E$ is called $\mu$-stable (resp $\mu$ semi-stable)if for any subsheaf $L$ of $E$ with rank $rank L\leq rank E$,

$$\frac{deg L}{rank L}<\mu\frac{deg E}{rank E}$$ $$\text{resp}\leq$$

Theorem (Tian 92)If $X$ is a Kahler-Einstein manifold with positive scalar curvature and $Pic(X)\cong \mathbb Z$ then the tangent bundle is $\mu$-semi-stable where here $\mu=\frac{n}{n+1}$.

Moreover if $X$ has no holomorphic vector field then $TX$ is $\frac{n}{n+1}$-stable.

Theorem: Tian proved that if $TX$ be $\frac{n}{n+1}$-stable, then $TX$ is strongly stable

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The first counterexample to your statement was given by K. Fujita in "Examples of K-unstable Fano manifolds with the Picard number one" available at arXiv:1508.04290.

There are examples of Mukai-Umerura threefolds with Picard rank one which are not K-polystable, the point of the conjecture you mention is that these threefolds are actually K-semistable (as discussed in the introduction to Fujita's paper).

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