# nef vs. 1-nef vector bundles

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 $$X$$?

[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

• We have 2-nef bundle, 3-nef bundle, ... $n$-nef bundle for any positive integer $n$ see Ugo Bruzzo, Beatriz Graña Otero, Metrics on semistable and numerically effective Higgs bundles, J. reine ang. Math. 612 (2007) 59-79 . I think when a VB is stable in the sense of Mumford, 1-nef bundle=nef bundle – user21574 Dec 8 '17 at 18:53
• Yes, in general I can consider $t$-nef bundles for $t\in\{1,...,\min\{d,r\}\}$ on $X$; $t$-nef is $s$-nef for $s\in\{1,...,t\}$; nef bundles are semistables in Mumford sense (it's a corollary of [2] theorem 2.18), and it turns out that $1$-nef bundles are semistables. But all this does not answer to my question! – Armando j18eos Dec 9 '17 at 8:18
• Interesting comment (with correct citation, I was careless to give the correct citation, sorry), First of all, that singular hermitian metric is not well defined for Vector bundles in general. You may assume that $h$ is negatively curved in the sense of Griffiths then you can define such singular hermitian metric on a VB .but I will think about it more. But I am not an expert, you may as Jean Pierre Demailly directly by email see also mathoverflow.net/questions/238825/… – user21574 Dec 9 '17 at 8:23
• Bruzzo with Biswas, Graña Otero, Gurjar, Hernández Ruipérez and others, extended the notions of nefness and $t$-nefness to Higgs bundles, respectively, on smooth projective varieties over algebrically closed fields of characteristic $0$ (H-nef) and connected, compact Kähler manifolds ($t$-H-nef). They proved that ($t$-)nef implies ($t$-)H-nef but the vice versa does not hold. – Armando j18eos Dec 9 '17 at 8:27
• I was wrong: the nef bundles with dual nef (numerically flat bundles, for short nflat) are semistables!, of course, it turns out that the $1$-nflat bundles ($1$-nef bundles with dual $1$-nef) are semistables. – Armando j18eos Dec 10 '17 at 14:24

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.

[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