  We carried out CO(1-0) and CO(2-1) observations of 21 different regions in
the vicinity of M86, NGC4438, and along the 120 kpc-long, Ha-emitting
filamentary trail that connects them, aiming to test whether molecular gas can
survive to be transferred from a spiral to an elliptical galaxy in Virgo's
10^7K intracluster medium (ICM). We targeted Ha-emitting regions that could be
associated with the interface between cold molecular clouds and the hot ionized
ICM. The data, obtained with the 30m telescope of the Institut de
Radioastronomie Millimetrique, led to the detection of molecular gas close to
M86. CO gas with a recession velocity that is similar to that of the stars,
-265 km/s, and with a corresponding H2 mass of 2*10^7 M_sun, was detected ~10
kpc southeast of the nucleus of M86, near the peak of its HI emission. We argue
that it is possible for this molecular gas either to have formed in situ from
HI, or to have been stripped from NGC4438 directly in molecular form. In situ
formation is nonetheless negligible for the 7*10^6 M_sun of gas detected at
12:26:15.9+12:58:49, at ~10 kpc northeast of M86, where no (strong) HI emission
is present. This detection provides evidence for the survival of molecular gas
in filaments for timescales of ~100 Myr. An amount equivalent to 5*10^7 M_sun
of H2 gas that could be lost to the ICM or to neighboring galaxies was also
discovered in the tidal tail northwest of NGC4438. A scenario of gas being
alternatively brought to M86 from NGC4388 on its south was also examined but it
was considered unlikely due to the non detection of CO below or at the HI
stream velocities, 2000-2700 km/s.
