I give here Gang Tian strong stability notion in the theory of slope stability as an application.

Let $X$ be a Fano manifold i.e( $-K_X>0$ or $c_1(X)>0$)

The existence of Kahler-Einstein metric on Fano manifold gives us the stability of tangent bundle $TX$. But the converse does not holds true always.

Example : Take $\Sigma_2$ by blowing up of $\mathbb CP^2$ at two point. Then $T\Sigma_2$ is stable but $\Sigma_2$ does not admit Kahler-Einstein metric since the Lie algebra of holomorphic vector fields is not reductive (as complexification of compact Lie gorup)due to Matsushima-Lichnerowicz.

Now Tian introduced a stronger notion of stability around 90 and called it strong stability 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: 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.

Theorem:(Tian)Suppose that $X$ admits a Kahler-Einstein metric $g$ with $Ric(g)=\omega_g$ where $\omega_g$ is the Kahler form of $g$. Then there is a natural Hermitian Yang-Mills metric $g_E$ on the extension of $E$ of $TX$ by the trivial sheaf and with the extension class $C_1(X)$. In particular the extension of $TX$ is semi-stable. Moreover if the extension is irreducible then it is strongly stable

I give here Gang Tian strong stability notion in the theory of slope stability as an application.

Let $X$ be a Fano manifold i.e( $-K_X>0$ or $c_1(X)>0$)

The existence of Kahler-Einstein metric on Fano manifold gives us the stability of tangent bundle $TX$. But the converse does not holds true always.

Example : Take $\Sigma_2$ by blowing up of $\mathbb CP^2$ at two point. Then $T\Sigma_2$ is stable but $\Sigma_2$ does not admit Kahler-Einstein metric since the Lie algebra of holomorphic vector fields is not reductive (as complexification of compact Lie gorup)due to Matsushima-Lichnerowicz.

Now Tian introduced a stronger notion of stability around 90 and called it strong stability 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: 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.

Theorem:(Tian)Suppose that $X$ admits a Kahler-Einstein metric $g$ with $Ric(g)=\omega_g$ where $\omega_g$ is the Kahler form of $g$. Then there is a natural Hermitian Yang-Mills metric $g_E$ on the extension of $E$ of $TX$ by the trivial sheaf and with the extension class $c_1(X)$. In particular the extension of $TX$ is semi-stable. Moreover if the extension is irreducible then it is strongly stable