When is a holomorphic tangent bundle stable?

Question

Hello, I was thinking about the stability condition in terms of Mumford and I have a question:

If $S$ is a compact Kahler surface (complex 2D), when is the holomorphic tangent bundle $T^{1,0}S$ stable?

When $S$ is Kahler-Einstein, $T^{1,0} S$ is stable by the Donaldson-Uhlenbeck-Yau theorem. But Kahler-Einstein examples are rather restrictive as it requires the hermitian metric $h$ on $TS$ to be induced by the riemannian metric $g$. Would there be more examples other than Kahler-Einstein with stable holomorphic tangent bundles?

Thank you in advance.

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The following has been added to clarify some points.

Appendix 1: What I meant by Kahler-Einstein being "restrictive".

Hermitian-Yang-Mills connections or equivalently Hermite-Einstein metrics can be thought as a generalization of Kahler-Einstein metrics. $T^{1,0}S$ admits a Hermite-Einstein metric with the hermitian structure induced from $(S, J, g)$ when $S$ is a Kahler-Einstein. However, in principle $T^{1,0}S$ can admit a Hermite-Einstein metric even when $S$ is not Kahler-Einstein. In this case, the hermitian structure on $T^{1,0}S$ will be different from the one naturally induced from $(S, J, g)$.

Appendix 2: What I meant by stable bundle.

The stability condition introduced by Mumford is the following.

A vector bundle $V$ is stable if for all coherent sub-sheaves $U$ $$\frac{\deg (U)}{\mathrm{rank} (U)}<\frac{\deg (V)}{\mathrm{rank} (V)}.$$

Here the degree is computed using the Kahler form $\omega$ (polarization). By the Donaldson-Uhlenbeck-Yau theorem, on Kahler manifolds stability is equivalent to $V$ admitting a Hermitian-Yang-Mills connection. This theorem was later generalized by Li and Yau assuming stability to non-Kahler manifolds.

Answer 1

What do you mean by stable? usually stable is with respect to a polarization $H$. If $K_X$ is ample and you choose $H=K_X$ then we know by Aubin-Yau's theorem that there exists a Kahler-Einstein metric. Thus, in this case, the existence of such a metric is not restrictive.

If you choose any ample line bundle $H$, then Donaldson has proved that if a holomorphic vector bundle $E$ over a compact Kahler manifold $M$ admits an approximate $\omega$-Einstein-Hermitian structure if and only if $E$ is $H$-semistable, where $\omega$ is a Kahler form in the class of $H$.

You can find many details in Kobayashi's book Differential geometry of complex vector bundles (official Mathematical Society of Japan new pdf version).