Positivity properties of the bundle of logarithmic tensors on compact Kahler manifolds

Let X be a compact Kahler manifold, endowed with an effective reduced divisor B = Sigma Y-k having simple normal crossing support. We consider a closed form of (1,1)-type alpha on X whose corresponding class {alpha} is nef, such that the class c(1) (K-X + B)+ {alpha} is an element of H-1,H-1(X, R) is pseudo-effective. A particular case of the first result we establish in this short note states the following. Let in be a positive integer, and let L be a line bundle on X, such that there exists a generically injective morphism L -> circle times T-m(X)star < B >, where we denote by T-X(star) < B > the logarithmic cotangent bundle associated to the pair (X, B). Then for any Kahler class {omega} on X, we have the inequality integral(x)c1(L)boolean AND{omega}(n-1) <= m integral(x) (c1 (K-X + B) + {alpha}) boolean AND {omega}(n-1) 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 circle times(m) T-X star < B > by L. Then its degree on a generic complete intersection curve C subset of X is bounded from below by As a consequence, we obtain a new proof of one of the main results of our previous work [F. Campana and M. Paun, Orbifold generic semi-positivity: an application to families of canonically polarized manifolds, Ann. Inst. Fourier (Grenoble) 65 (2015), 835-861].