Let
$X$
be a compact Kähler manifold, endowed with an effective reduced divisor
$B=\sum Y_{k}$
having simple normal crossing support. We consider a closed form of
$(1,1)$
-type
$\unicode[STIX]{x1D6FC}$
on
$X$
whose corresponding class
$\{\unicode[STIX]{x1D6FC}\}$
is nef, such that the class
$c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\}\in H^{1,1}(X,\mathbb{R})$
is pseudo-effective. A particular case of the first result we establish in this short note states the following. Let
$m$
be a positive integer, and let
$L$
be a line bundle on
$X$
, such that there exists a generically injective morphism
$L\rightarrow \bigotimes ^{m}T_{X}^{\star }\langle B\rangle$
, where we denote by
$T_{X}^{\star }\langle B\rangle$
the logarithmic cotangent bundle associated to the pair
$(X,B)$
. Then for any Kähler class
$\{\unicode[STIX]{x1D714}\}$
on
$X$
, we have the inequality
$$\begin{eqnarray}\displaystyle \int _{X}c_{1}(L)\wedge \{\unicode[STIX]{x1D714}\}^{n-1}\leqslant m\int _{X}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})\wedge \{\unicode[STIX]{x1D714}\}^{n-1}.\end{eqnarray}$$
If
$X$
is projective, then this result gives a generalization of a criterion due to Y. Miyaoka, concerning the generic semi-positivity: under the hypothesis above, let
$Q$
be the quotient of
$\bigotimes ^{m}T_{X}^{\star }\langle B\rangle$
by
$L$
. Then its degree on a generic complete intersection curve
$C\subset X$
is bounded from below by
$$\begin{eqnarray}\displaystyle \biggl(\frac{n^{m}-1}{n-1}-m\biggr)\int _{C}(c_{1}(K_{X}+B)+\{\unicode[STIX]{x1D6FC}\})-\frac{n^{m}-1}{n-1}\int _{C}\unicode[STIX]{x1D6FC}.\end{eqnarray}$$
As a consequence, we obtain a new proof of one of the main results of our previous work [F. Campana and M. Păun,
Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble)
65 (2015), 835–861].