We have made hydro/N-body simulations with and without star formation to shed some light on the conditions under which a central kinematically cold stellar component ( characterized by a velocity dispersion drop or \sigma -drop ) could be created in a hot medium ( e.g .
a bulge ) and survive enough time to be observed .
We found that the timescale for a \sigma -drop formation could be short ( less than 500 Myr ) whereas its lifetime could be long ( more than 1 Gyr ) provided that the central region is continuously or regularly fed by fresh gas which leads to a continuous star formation activity .
Star formation in the central region , even at a low rate as 1 M _ { \sun } \mbox { yr } ^ { -1 } , is mandatory to sustain a permanent \sigma -drop by replacing heated particles by new low- \sigma ones .
We moreover show that as soon as star formation is switched off , the \sigma -drop begins to disappear .