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

By [1] definition 3.1.2:

> A line bundle $L$ over $X$ 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. $E$ is *nef* if the tautological bundle $\mathcal{O}_{\mathbb{P}(E)}(1)$ is *nef*.

By [1] definition 3.1.3:

> $E$ is *$1$-nef* if for any $\epsilon>0$ there exists a Hermitian metric $h_{\epsilon}$ on $E$ such that $\Omega_{h_{\epsilon}}(E)\geq-\epsilon\omega$.

By [1] proposition 3.2.4, the $1$-nef bundles $E$ over $X$ are nef; but the inverse is unknown in general; excepted for:

 1. $d=1$ (i.e. algebraic curves), see [1] theorem 3.3.1;
 2. on toric and Abelian varieties $E\otimes\det E$ is $1$-nef;
 3. tangent bundle $TX$ of $X$, where it is nef and $d\in\{2,3\}$;

(2) is justified in [1] at page 113, (3) follows by [2] theorems 6.1, 7.1 and [1] proposition 3.2.4.

**Question:** Are there other examples of manifolds over which the nef bundles are $1$-nef? Or is there an example of nef not $1$-nef bundle over some manifold $X$?

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[1] M. A. A. De Cataldo - *Singular Hermitian metrics on vector bundles*, J. reine ang. Math. **502** (1998) 93-122

[2] J.-P. Demailly, T. Peternell, M. Schneider - _Compact complex manifolds with numerically effective tangent bundles_, J. Algebraic Geom. **3** (1994) 295-345