Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.

By [1] definition 1.2:

A line bundle $L$ on $X$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth Hermitian metric $h_{\epsilon}$ on $L$ such that \begin{equation*} \Omega_{h_{\epsilon}}(L)\geq-\epsilon\omega; \end{equation*} that is the curvature form $\Omega_{h_{\epsilon}}(L)$ of the Chern connection on $L$ (with respect to $h_{\epsilon}$) can have an arbitrary negative part.

By [1] definition 1.9:

$E$ is nef if the tautological bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is nef. $E$ is numerically flat (nflat, for short) if $E$ and $E^{\vee}$ are both nef.

In the proof of theorem 1.18, one assumes $E$ nflat; let $F$ be a reflexive subsheaf of $E$ of minimal rank $p$ of degree $0$, one proves that $\det F$ is a line bundle and by [1] lemma 1.20 $F$ is a subbundle of $E$. By assumption, $F$ is a ($\omega$-)stable subbundle of $E$.

The authors state that $F^{\vee}$ is a locally free quotient sheaf of $E^{\vee}$!

Question: Why does this hold?

I understand that $\left(E_{\displaystyle/F}\right)^{\vee}\equiv\mathcal{Q}^{\vee}$ is a reflexive subsheaf of $E^{\vee}$ (cfr. [2] proposition V.5.18 or this answer), on an open subset $U$ of $X$ it is locally free, so $\mathcal{E}xt^1_{\mathcal{O}_X}\left(\mathcal{Q},\mathcal{O}_X\right)_{|U}=0$ and $F^{\vee}_{|U}$ is a locally free quotient sheaf of $E^{\vee}_{|U}$.

Remark: For the case of complex curves and Kähler surfaces, the previous statement holds by [2] corollary V.5.20; because, without change the names, $\textrm{codim}(X\setminus U)\geq3$.

[1] J.-P. Demailly, T. Peternell, M. Schneider - Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994) 295-345

[2] S. Kobayashi (1987) Differential Geometry of Complex Vector Bundles, Iwanami Shoten Publishers and Princeton University Press

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.

By [1] definition 1.2:

A line bundle $L$ on $X$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth Hermitian metric $h_{\epsilon}$ on $L$ such that \begin{equation*} \Omega_{h_{\epsilon}}(L)\geq-\epsilon\omega; \end{equation*} that is the curvature form $\Omega_{h_{\epsilon}}(L)$ of the Chern connection on $L$ (with respect to $h_{\epsilon}$) can have an arbitrary negative part.

By [1] definition 1.9:

$E$ is nef if the tautological bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is nef. $E$ is numerically flat (nflat, for short) if $E$ and $E^{\vee}$ are both nef.

In the proof of theorem 1.18, one assumes $E$ nflat; let $F$ be a reflexive subsheaf of $E$ of minimal rank $p$ of degree $0$, one proves that $\det F$ is a line bundle and by [1] lemma 1.20 $F$ is a subbundle of $E$. By assumption, $F$ is a ($\omega$-)stable subbundle of $E$.

The authors state that $F^{\vee}$ is a locally free quotient sheaf of $E^{\vee}$!

Question: Why does this hold?

I understand that $\left(E_{\displaystyle/F}\right)^{\vee}\equiv\mathcal{Q}^{\vee}$ is a reflexive subsheaf of $E^{\vee}$ (cfr. [2] proposition V.5.18 or this answer), on an open subset $U$ of $X$ it is locally free, so $\mathcal{E}xt^1_{\mathcal{O}_X}\left(\mathcal{Q},\mathcal{O}_X\right)_{|U}=0$ and $F^{\vee}_{|U}$ is a locally free quotient sheaf of $E^{\vee}_{|U}$.

Remark: For the cases of complex curves and Kähler surfaces, the previous statement holds by [2] corollary V.5.20; because, without change the names, $\textrm{codim}(X\setminus U)\geq3$.

[1] J.-P. Demailly, T. Peternell, M. Schneider - Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994) 295-345

[2] S. Kobayashi (1987) Differential Geometry of Complex Vector Bundles, Iwanami Shoten Publishers and Princeton University Press

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.

By [1] definition 1.2:

A line bundle $L$ on $X$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth Hermitian metric $h_{\epsilon}$ on $L$ such that \begin{equation*} \Omega_{h_{\epsilon}}(L)\geq-\epsilon\omega; \end{equation*} that is the curvature form $\Omega_{h_{\epsilon}}(L)$ of the Chern connection on $L$ (with respect to $h_{\epsilon}$) can have an arbitrary negative part.

By [1] definition 1.9:

$E$ is nef if the tautological bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is nef. $E$ is numerically flat (nflat, for short) if $E$ and $E^{\vee}$ are both nef.

In the proof of theorem 1.18, one assumes $E$ nflat; let $F$ be a reflexive subsheaf of $E$ of minimal rank of degree $0$, so $F$ is a ($\omega$-)stable subbundle of $E$ (cfr. [1] lemma 1.20).

The author state that $F^{\vee}$ is a locally free quotient sheaf of $E^{\vee}$!

Question: Why does this hold?

I understand that $\left(E_{\displaystyle/F}\right)^{\vee}\equiv\mathcal{Q}^{\vee}$ is a coherent subsheaf of $E^{\vee}$, on an open subset $U$ of $X$ it is locally free, so $\mathcal{E}xt^1_{\mathcal{O}_X}\left(\mathcal{Q},\mathcal{O}_X\right)_{|U}=0$ and $F^{\vee}_{|U}$ is locally free.

Remark: For the case of complex curves and Kähler surfaces, the previous statement holds by [2] corollary V.5.20; because, without change the names, $\textrm{codim}(X\setminus U)\geq3$.

[1] J.-P. Demailly, T. Peternell, M. Schneider - Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994) 295-345

[2] S. Kobayashi (1987) Differential Geometry of Complex Vector Bundles, Iwanami Shoten Publishers and Princeton University Press

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.

By [1] definition 1.2:

A line bundle $L$ on $X$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth Hermitian metric $h_{\epsilon}$ on $L$ such that \begin{equation*} \Omega_{h_{\epsilon}}(L)\geq-\epsilon\omega; \end{equation*} that is the curvature form $\Omega_{h_{\epsilon}}(L)$ of the Chern connection on $L$ (with respect to $h_{\epsilon}$) can have an arbitrary negative part.

By [1] definition 1.9:

$E$ is nef if the tautological bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is nef. $E$ is numerically flat (nflat, for short) if $E$ and $E^{\vee}$ are both nef.

In the proof of theorem 1.18, one assumes $E$ nflat; let $F$ be a reflexive subsheaf of $E$ of minimal rank $p$ of degree $0$, one proves that $\det F$ is a line bundle and by [1] lemma 1.20 $F$ is a subbundle of $E$. By assumption, $F$ is a ($\omega$-)stable subbundle of $E$.

The authors state that $F^{\vee}$ is a locally free quotient sheaf of $E^{\vee}$!

Question: Why does this hold?

I understand that $\left(E_{\displaystyle/F}\right)^{\vee}\equiv\mathcal{Q}^{\vee}$ is a reflexive subsheaf of $E^{\vee}$ (cfr. [2] proposition V.5.18 or this answer), on an open subset $U$ of $X$ it is locally free, so $\mathcal{E}xt^1_{\mathcal{O}_X}\left(\mathcal{Q},\mathcal{O}_X\right)_{|U}=0$ and $F^{\vee}_{|U}$ is a locally free quotient sheaf of $E^{\vee}_{|U}$.

Remark: For the case of complex curves and Kähler surfaces, the previous statement holds by [2] corollary V.5.20; because, without change the names, $\textrm{codim}(X\setminus U)\geq3$.

[1] J.-P. Demailly, T. Peternell, M. Schneider - Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994) 295-345

[2] S. Kobayashi (1987) Differential Geometry of Complex Vector Bundles, Iwanami Shoten Publishers and Princeton University Press

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.

By [1] definition 1.2:

A line bundle $L$ on $X$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth Hermitian metric $h_{\epsilon}$ on $L$ such that \begin{equation*} \Omega_{h_{\epsilon}}(L)\geq-\epsilon\omega; \end{equation*} that is the curvature form $\Omega_{h_{\epsilon}}(L)$ of the Chern connection on $L$ (with respect to $h_{\epsilon}$) can have an arbitrary negative part.

By [1] definition 1.9:

$E$ is nef if the tautological bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is nef. $E$ is numerically flat (nflat, for short) if $E$ and $E^{\vee}$ are both nef.

In the proof of theorem 1.18, one assumes $E$ nflat; let $F$ be a reflexive subsheaf of $E$ of minimal rank of degree $0$, so $F$ is a ($\omega$-)stable subbundle of $E$ (cfr. [1] lemma 1.20).

The author state that $F^{\vee}$ is a locally free quotient sheaf of $E^{\vee}$!

Question: Why is this hold?

I understand that $\left(E_{\displaystyle/F}\right)^{\vee}\equiv\mathcal{Q}^{\vee}$ is a coherent subsheaf of $E^{\vee}$, on an open subset $U$ of $X$ it is locally free, so $\mathcal{E}xt^1_{\mathcal{O}_X}\left(\mathcal{Q},\mathcal{O}_X\right)_{|U}=0$ and $F^{\vee}_{|U}$ is locally free.

Remark: For the case of complex curves and Kähler surfaces, the previous statement holds by [2] corollary V.5.20; because $\textrm{codim}X\setminus U\geq3$.

[1] J.-P. Demailly, T. Peternell, M. Schneider - Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994) 295-345

[2] S. Kobayashi (1987) Differential Geometry of Complex Vector Bundles, Iwanami Shoten Publishers and Princeton University Press

Let $X$ be a compact connected Kähler manifold, of dimension $d\geq3$, with Hermitian metric $\omega$; let $E$ be a vector bundle on $X$ of rank $r\geq2$.

By [1] definition 1.2:

A line bundle $L$ on $X$ is said numerically effective (nef, for short) if for any $\epsilon>0$ there exists a smooth Hermitian metric $h_{\epsilon}$ on $L$ such that \begin{equation*} \Omega_{h_{\epsilon}}(L)\geq-\epsilon\omega; \end{equation*} that is the curvature form $\Omega_{h_{\epsilon}}(L)$ of the Chern connection on $L$ (with respect to $h_{\epsilon}$) can have an arbitrary negative part.

By [1] definition 1.9:

$E$ is nef if the tautological bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is nef. $E$ is numerically flat (nflat, for short) if $E$ and $E^{\vee}$ are both nef.

In the proof of theorem 1.18, one assumes $E$ nflat; let $F$ be a reflexive subsheaf of $E$ of minimal rank of degree $0$, so $F$ is a ($\omega$-)stable subbundle of $E$ (cfr. [1] lemma 1.20).

The author state that $F^{\vee}$ is a locally free quotient sheaf of $E^{\vee}$!

Question: Why does this hold?

I understand that $\left(E_{\displaystyle/F}\right)^{\vee}\equiv\mathcal{Q}^{\vee}$ is a coherent subsheaf of $E^{\vee}$, on an open subset $U$ of $X$ it is locally free, so $\mathcal{E}xt^1_{\mathcal{O}_X}\left(\mathcal{Q},\mathcal{O}_X\right)_{|U}=0$ and $F^{\vee}_{|U}$ is locally free.

Remark: For the case of complex curves and Kähler surfaces, the previous statement holds by [2] corollary V.5.20; because, without change the names, $\textrm{codim}(X\setminus U)\geq3$.

[1] J.-P. Demailly, T. Peternell, M. Schneider - Compact complex manifolds with numerically effective tangent bundles, J. Algebraic Geom. 3 (1994) 295-345

[2] S. Kobayashi (1987) Differential Geometry of Complex Vector Bundles, Iwanami Shoten Publishers and Princeton University Press