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In this paper we study arakelovian heights with respect to line bundles endowed with some logarithmically singular metrics. After setting some quasi-plurisubharmonicity properties of these kind of metrics, we show the corresponding heights satisfy the usual finiteness and universal lower bound properties. Therefore we simultaneously generalize a lemma due to Faltings in his proof of Mordell's conjecture and some results of Bost-Gillet-Soulé. The main theorem is a diophantine inequality, providing a vast generalization of Liouville's (trivial) inequality, but replacing "algebraic number" by "algebraic divisor", "rational point" by "effective cycle", etc. In some sense, arakelovian heights w.r.t. logarithmically singular metrics give a way of defining Weil distances from a cycle to a divisor. We propose as an open question a higher dimensional analog to Roth's theorem. For this we introduce complete Kähler-Einstein metrics on quasi-projective manifolds. We are somehow inspired by the equidimensional Nevanlinna theory of Carlson and Griffiths as well as Vojta's conjectures. We give another example of the techniques to deal with such metrics: a generalization of Zhang's inequality for heights of generic sequences of points. |