# Applications of Slope Stability

Ross and Thomas developed slope-stability of $(X,L)$ where $X$ is an $L$-polarised variety and $L$ is an ample line bundle, as an obstruction to K-stability of $(X,L)$.

DISCLAIMER: (Forgive me if I don't define what these are, but for the purpose of the question if you do not know them well already is not going to help you. Moreover the definition is rather technical. Also, I am not an expert in these topics, so I expect to say a couple of things wrong without being aware of it)

K-stability is an obstruction to the existence of Kahler-Einstein metrics on $X$. Therefore slope-stability can be seen as a tool to decide when a Fano variety does not admit a Kahler-Einstein metric:

Kahler einstein $\Rightarrow$ K-stable $\Rightarrow$ slope-stable.

The last arrow is not strict, i.e. there are slope-stable $(X,L)$ which are not K-stable. On the other hand, computing slope-stability is much easier than computing K-stability.

I know that K-stability has other applications, for instance if $(X,\mathcal{O}(-mK_X)),\ m\in \mathbb{Z}_{>0}$ satisfies certain conditions (including being Fano) and K-semistability, then the singularities of $X$ are log terminal by a result by Odaka in Annals. Therefore K-stability is interesting not only within the Kahler-Einstein problem.

I was wondering if there is any other applications of slope-stability other than as an obstruction to K-stability.

(answers considering the log setting are also welcomed)

There are quite a lot of applications to classical algebraic geometry.

One application (which appears in the original paper of Ross and Thomas, Theorem 3.9) is to obstruct other more classical notions of stability, such as Hilbert or Chow stability.

More precisely, they show that if a polarized manifold is slope unstable, then it cannot be asymptotically Hilbert semistable or asymptotically Chow semistable.

Apart from the word "asymptotically", these were classical notions of stability in algebraic geometry, see e.g. Mumford's classical paper.

This is applied for example to the case of a selfproduct of a curve $C$ of genus $5$ or more by Ross, where he shows that there are polarizations on $C\times C$ which are not asymptotically Hilbert or Chow semistable.

Another application (again in the original paper of Ross-Thomas, Theorem 7.16) is in the case of curves, where they use slope stability to reprove a result of Mumford, that smooth curves of genus at least $1$ are asymptotically Chow stable.

This was later extended to the case of singular (weighted pointed) nodal curves by J.Li and X.Wang.

• Acutally, the work on nodal curves by Li and Wang uses a different notion of "slope". – JRoss Oct 19 '12 at 18:55

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

• Moreover Tian with Ding showed that cubic surface in $\mathbb CP^3$ has a Kähler-Einstein orbifold metric only if it is semistable in the sense of Takemoto-Mumford. Ding, Wei Yue, Tian, Gang, Kähler-Einstein metrics and the generalized Futaki invariant. Invent. Math. 110 (1992), no. 2, 315–335. – user21574 Jul 23 '17 at 16:38
• Note that to check K-semistability it is sufficient to consider only test configurations that have smooth total space and central fibre that is a reduced simple normal divisor. This gives a connection between Mumford's semi-stable reduction theory and Kahler-Einstein geometry. See Arezzo et al paper. This result later refined by Chi Li et al as special test configuration – user21574 Jul 31 '17 at 2:49