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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)

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up vote 1 down vote accepted

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.

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This is great! Thanks a lot :) – Jesus Martinez Garcia Oct 16 '12 at 9:20
Acutally, the work on nodal curves by Li and Wang uses a different notion of "slope". – JRoss Oct 19 '12 at 18:55

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