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 $\textrm{codim}X\setminus U\geq3$.

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[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