Semi-stability of $S^n\Omega_S$ with respect to $K_S$

Question

Let $S$ be a minimal compact complex surface of general type with ample canonical class $K_S$. In [1, Theorem 3] the following result is stated:

Theorem. Every symmetric power $S^n \Omega_S$ of the cotangent bundle $\Omega_S$ is stable with respect to $K_S$, unless $S$ is uniformized by the bi-disk.

I understand why the assumption about the universal covering is necessary: indeed, if for instance $S=C_1\times C_2$, with $C_i$ smooth curve of genus $g \geq 2$, then $\Omega_S$ is the direct sum of two line bundles; thus, every symmetric power is also reducible as a sum of line bundles, in particular, it is non-stable.

However, if I made the computations correctly, in this example all direct summands in $S^n \Omega_S$ have the same slope with respect to $K_S$, hence the symmetric powers of $\Omega_S$ are $K_S$-semistable.

I wonder if this is true in general; I asked some people and I was told that this should be in fact a result of Bogomolov, but I was not given a precise reference, and I was unable to locate one. So, let me ask the

Question. If $S$ is a compact complex surface as above, is it true that every symmetric power $S^n\Omega_S$ is $K_S$-semistable? If so, what is a reference?

References.

[1] Lu, Steven Shin-Yi, On hyperbolicity and the Green-Griffiths conjecture for surfaces, Noguchi, J. (ed.) et al., Geometric complex analysis. Proceedings of the conference held at the 3rd International Research Institute of the Mathematical Society of Japan, Hayama, March 19-29, 1995. Singapore: World Scientific. 401-408 (1996). ZBL0941.32024.

Answer 1

Ciao Francesco!

The answer to your question is yes, and the work of Bogomolov you are looking for about semistability of the tangent space (for minimal surfaces of general type indeed, no need of ampleness of the canonical bundle) is

"Holomorphic tensors and vector bundles on projective varieties", Math. USSR Izvestija 13/3 (1979) 499-555.

But let me give you a more general framework.

Given a Hermitian holomorphic vector bundle $(E,h)$ over a compact Kähler manifold $(X,\omega)$, if $(E,h)\to (X,\omega)$ is Hermite-Einstein then its dual as well as its symmetric powers (with the induced Hermitian structures) also are Hermite-Einstein.

Next, if $(E,h)\to (X,\omega)$ is Hermite-Einstein, then $E\to X$ is $[\omega]$-semistable (polystable, indeeed). This is the "easy" direction of the Kobayashi-Hitchin correspondence.

If a projective manifold $X$ has ample canonical bundle, then it admits a Kähler-Einstein metric $\omega$ of negative Einstein constant, i.e. a Kähler metric such that $\operatorname{Ric}(\omega)=-\omega$ (by the Aubin-Yau Theorem).

In particular $(T_X,\omega)\to (X,\omega)$ is Hermite-Einstein, and then so does any symmetric power of the cotangent bundle (with the induced Hermitian structure). This makes these vector bundles $[\omega]$-semistable, as we saw above.

Finally, in this case, begin $[\omega]$-semistable means that these vector bundle are $K_X$-semistable, since $[\omega]=[-\operatorname{Ric}(\omega)]=c_1(K_X)$.

You can see all this and much more on S. Kobayashi "Differential geometry of complex vector bundles".

P.S. For quite recent advances about semistability of the tangent sheaf of possibly singular varieties (even in the logarithmic setting), you might want to take a look to this paper by H. Guenancia.