nef vs. 1-nef vector bundles

Question

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

Answer 1

Let $(X,\omega)$ be a compact Kähler manifold of dimension $d\geq2$; let $E$ be a nef vector bundle of rank $r\geq2$ such that $\displaystyle\deg(E)=\int_Xc_1(E)\wedge\omega^{d-1}=0$. $E$ is a $1$-nflat vector bundle.

Proof. By [DPS, propositions 1.14.i and 1.15.i] $\det(E)$ is a ($1$-)nef line bundle; by [BGO2, remark 3.2.ii and lemma 3.13] $\det(E)$ is a ($1$-)nflat line bundle; because \begin{equation*} E^{\vee}\cong\bigwedge^{r-1}E\otimes(\det(E))^{\vee} \end{equation*} $E^{\vee}$ is a nef vector bundle. By definition $E$ is a nflat vector bundle.

By [DPS, theorem 3.2] $E$ has a filtration whose quotients are locally free and flat, by [BGO2, theorem 3.16] this is equivalent to say that $E$ is $1$-nflat. $\Box$

Remark. Using the notion of Numerically Flat Higgs bundle (H-nflat, for short) introduced in [BGO1], and $1$-Numerically Flat Higgs bundle ($1$-H-nflat, for short) introduced in [BGO2], and applying [BC, theorems 3.2 and 5.2] indeed of [DPS, theorem 3.2], one proves the following (partial) generalization to Higgs bundles case.

Let $(X,H)$ be a smooth complex projetive variety of dimension $d\geq2$, with a polarization $H$; let $\mathfrak{E}=(E,\phi)$ be a H-nef Higgs bundle of rank $r\geq2$ such that $\displaystyle\deg(E)=c_1(E)\cdot H^{d-1}=0$ and $c_2(E)\cdot H^{d-2}=0$. $\mathfrak{E}$ is a $1$-H-nflat Higgs bundle.

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[BC] U. Bruzzo, A. Capasso, Filtrations of numerically flat Higgs bundles and curve semistable Higgs bundles on Calabi-Yau manifolds, arXiv:1904.10069 [math.AG]

[BGO1] U. Bruzzo and B. Graña Otero, Numerically Flat Higgs bundles, Commun. Contemp. Math. 9 (2007) 437-446

[BGO2] U. Bruzzo and B. Graña Otero, Metrics on semistable and numerically effective Higgs bundles, J. reine ang. Math. 612 (2007) 59-79.

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