We report results from our mini-survey for molecular hydrogen in eight high redshift damped Lyman-alpha (DLA) systems using the ESO Ultra-violet and Visible Spectrograph on the VLT. In addition, we investigate two systems using ESO public data. We include in the sample the only system where H2 was previously detected and studied at high-spectral resolution. Altogether our sample consists of eleven absorbers with 1.85<z_abs<3.4. We confirm the presence of H2 in the z_abs = 2.3377, metal-poor ([Si/H] = -1.20) system toward PKS 1232+082. The derived molecular fraction, f = 2N(H2)/(2N(H2)+N(HI)) = 4 10-4, is two orders of magnitude less than what has been claimed previously from low-resolution data. The physical conditions within the cloud can be constrained directly from observation. The kinetic temperature and particle density are in the ranges, respectively, 100<T<300 K and 30<nH<50 cm-3. In addition, UV pumping is of the same order of magnitude than in our Galaxy. The upper limits on the molecular fraction derived in nine of the systems are in the range 1.2 10-7 - 1.6 10-5. The molecular abundance in a few DLA systems (and in particular in the two systems where H2 is detected) is consistent with what is seen in the Magellanic clouds. But most of the DLA measurements are well below these values. This is probably partly due to small amounts of dust and/or high UV flux. We argue however that the lack of molecules is a direct consequence of high kinetic temperature (T>3000 K) implying a low formation rate of H2 onto dust grains. Therefore, most of the DLA systems arise in warm and diffuse neutral gas. (Abridged)
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