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 H _ { 2 } 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 H _ { 2 } in the z _ { abs } = 2.3377 , metal-poor ( [ Si/H ] = - 1.20 ) , system toward PKS 1232+082 .
The derived molecular fraction , f = 2 N ( H _ { 2 } ) / ( 2 N ( H _ { 2 } ) + N ( H i ) ) = 4 \times 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 < n _ { \sc H } < 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 \times 10 ^ { -7 } - 1.6 \times 10 ^ { -5 } .
There is no evidence in this sample for any correlation between H _ { 2 } abundance and relative heavy element depletion into dust grains .
This should be investigated using a larger sample however .
The molecular abundance in a few DLA systems ( and in particular in the two systems where H _ { 2 } 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 H _ { 2 } onto dust grains .
Therefore , most of the DLA systems arise in warm and diffuse neutral gas .