Detailed high resolution observations of the innermost central parts of nearby galaxies have revealed the presence of supermassive black holes 1-4 . These black holes may interact with 1 their host galaxies by means of 'feedback' in the form of energy and material jets; this feedback affects the evolution of the host and gives rise to the observed relations between the black hole and the host 5 . Here we report observations of the ultraviolet emissions of massive early-type galaxies. We derive an empirical relation for a critical black-hole mass (as a function of velocity dispersion) above which the outflows from these black holes suppress star formation in their hosts by heating and expelling all available cold gas. Supermassive black holes are negligible in mass compared to their hosts but nevertheless seem to play a critical role in the star formation history of galaxies.The near-UV (NUV) detector of the Galaxy Evolution Explorer satellite GALEX 6 covers a range in wavelength between 1771 and 2831Å and is extremely sensitive to young stellar populations. With it, we can detect small mass fractions of 1-3% of young stars formed within the last billion years 7, 8 . This high sensitivity allows us to trace ongoing residual star formation in present day early-type galaxies (ETGs). We select a volume-limited sample of visually inspected ETGs from the Sloan Digital Sky Survey 9 that have been observed by GALEX. We remove active galactic nucleus (AGN) candidates from our sample to avoid confusion between the observed UV emissions from the AGN and from young stars 1 .We use NUV − r colour as a probe of small amounts of recent star formation (RSF) and find that the fraction of ETGs that show signs of RSF is strongly correlated with the stellar velocity dispersion σ, (hereafter "RSF-σ relation"), but not with the luminosity of the galaxy (see Figure 1). The correlation with σ cannot be explained by a systematic trend of σ with internal extinction, 2 because such a trend would be exact opposite 11 to that observed.We use a semi-analytic model of galaxy evolution in the ΛCDM paradigm to interpret these results 2, 3, 14 (for model details see Supplements). Our simulation includes all the systematic processes that affect galaxy evolution and takes into account the different merger and star formation histories of galaxies 15 . When we run this simulation and investigate the predicted amount of recent star formation in low redshift ETGs, we find that all galaxies underwent enough star formation within the last billion years to be detected by the extremely sensitive GALEX NUV detector. Massive galaxies continued to gain cold gas as fuel for star formation from mergers with satellites and through the cooling of hot gas present in those galaxies.This overproduction of stars in massive galaxies is a known problem in models of galaxy evolution. In order to prevent star formation, a shutdown mechanism is required: cold gas must be heated up and expelled to prevent it from turning into stars. Feedback from supernovae (SNe) was o...