The HRS is a complete volume-limited sample of nearby objects including Virgo cluster and isolated objects. Using a recent compilation of HI and CO data we study the effects of the cluster on the molecular gas content of spiral galaxies. We first identify M* as the scaling variable that traces the total H2 mass of galaxies better. We show that, on average, HI-deficient galaxies are significantly offset from the M(H2) vs. M* relation for HI-normal galaxies. We use the M(H2) vs. M* scaling relation to define the H2-deficiency parameter. This parameter shows a weak and scattered relation with the HI-def, here taken as a proxy for galaxy interactions with the cluster environment. We also show that, as for the HI, the extent of the H2 disc decreases with increasing HI-deficiency. These results show that cluster galaxies have, on average, a lower H2 content than similar objects in the field. The slope of the H2-def vs. HI-def relation is less than 1, while the D(HI)/D(i) vs. HI-def relation is steeper than the D(CO)/D(i) vs. HI-def relation, thereby indicating that the H2 gas is removed less efficiently than the HI. This result can be understood if the HI is distributed on a flat disc more extended than the stellar disc, thus less anchored to the gravitational potential well of the galaxy than the H2. There is a clear trend between the NUV-i colour and H2-def, which suggests that H2 removal quenches the activity of star formation. This causes galaxies migrate from the blue cloud to the green valley and, eventually, to the red sequence. The total gas-consumption timescale of gas deficient cluster galaxies is comparable to that of isolated systems, and is significantly larger than the typical timescale for total gas removal in a ram pressure stripping process, thus suggesting that ram pressure, rather than starvation, is the dominant process driving the evolution of these cluster galaxies.
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