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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. Is $S$ 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.

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.

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. Is $S$ 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.

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. Is $S$ 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.

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 reducible as a sum of line bundles, in particular, it is non-stable.

However, if I made the computations correctly, 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. Is $S$ 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.

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. Is $S$ 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.