  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)
