Claire Voisin defined a variant of the Kobayashi-Eisenman
pseudo-volume form of a complex manifold X, by introducing the notion of holomorphic K-correspondance, in order to replace holomorphic maps between the unit polydisk D^n and X in the definition by them. I will study an adaptation of these constructions to the frame of logarithmic pairs (X, D), where X is a complex manifold, and D is a normal crossing divisor, the positive part of wich is reduced. I will define an intrinsic pseudo-volume form Phi_{X,D} for every logarithmic pair (X, D). I will prove on the one hand that Phi_{X,D} is generically non-degenerate if K_X(D) is ample and the positive part of D is simply normal crossing (this corresponds to the Griffiths and Kobayshi-Ochiai theorem in the standard case). On the other hand, I will show the vanishing of Phi_{X,D} for a large class of pairs with trivial logarithmic canonical bundle, which is an important step towards an analogous of the Kobayashi conjecture on measure-hyperbolicity. |

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