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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".

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.

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".

Let me finish by telling you that Tsuji proved in 1988 that more generally every minimal projective manifold of general type $X$ (i.e. $K_X$ big&nef) has tangent bundle which is $K_X$-semistable.

But now there is a problem: morally you would like to pass from the semistability of a vector bundle to the semistability of its symmetric (or, more generally tensor) powers. For stable or polystable vector bundle this is usually done via the (hard direction of the) Kobayashi-Hitchin correspondence, but if the bundle is only semistable but not polystable, you only have an "approximate" Hermite-Einstein structure, which is not enough to infer semistability of the tensor powers.

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".

Ciao Francesco!

In general 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 $T_X$ admits a Hermite-Einstein metric, and then so does any irreducible $\operatorname{GL}(T_X)$-representation, such as the symmetric powers of the cotangent bundle. This makes these vector bundles $[\omega]$-semistable, by the easy direction of the Kobayashi-Hitchin correspondence.

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. The work of Bogomolov you are looking for is

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

Moreover, you don't need $K_X$ to be ample, big&nef suffices. I'll expand later, sorry but I don't have time now.

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".

Let me finish by telling you that Tsuji proved in 1988 that more generally every minimal projective manifold of general type $X$ (i.e. $K_X$ big&nef) has tangent bundle which is $K_X$-semistable.

But now there is a problem: morally you would like to pass from the semistability of a vector bundle to the semistability of its symmetric (or, more generally tensor) powers. For stable or polystable vector bundle this is usually done via the (hard direction of the) Kobayashi-Hitchin correspondence, but if the bundle is only semistable but not polystable, you only have an "approximate" Hermite-Einstein structure, which is not enough to infer semistability of the tensor powers.