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](https://mathoverflow.net/questions/17277/dual-of-torsion-free-reflexive-coherent-sheaf/17317#17317)), 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, $\textrm{codim}(X\setminus U)\geq3$.




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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



Dual of a stable locally free subsheaf is a locally free quotient sheaf
If $F$ is a *subbundle* of $E$, then one has a natural surjection of vector bundles $E^\*\to F^\*\to 0$ induced by the restriction map. This is equivalent to saying the $F^\*$ is a quotient bundle of $E^\*$ (by the kernel of the map above, which is a vector subbundle of $E^\*$), or if you prefer, $\mathcal O\_X(F^\*)$ is a locally free quotient of $\mathcal O\_X(E^\*)$. 


