  [Abridged] We investigate trends between the recent star formation history
and black hole growth in galaxy bulges in the Sloan Digital Sky Survey (SDSS).
The galaxies lie at 0.01<z<0.07 where the fibre aperture covers only the
central 0.6-4.0kpc diameter of the galaxy. We find strong trends between black
hole growth, as measured by dust-attenuation-corrected OIII luminosity, and the
recent star formation history of the bulges. We conclude that our results
support the popular hypothesis for black hole growth occurring through gas
inflow into the central regions of galaxies, followed by a starburst and
triggering of the AGN. However, while this is a significant pathway for the
growth of black holes, it is not the dominant one in the present-day Universe.
More unspectacular processes are apparently responsible for the majority of
this growth.
  In order to arrive at these conclusions we have developed a set of new high
signal-to-noise ratio (SNR) optical spectral indicators, designed to allow a
detailed study of stellar populations which have undergone recent enhanced star
formation. Working in the rest-frame wavelength range 3750-4150AA, ideally
suited to many recent and ongoing spectroscopic surveys at low and high
redshift, the first two indices are equivalent to the previously well studied
4000AA break strength and Hdelta equivalent width. The primary advantage of
this new method is a greatly improved SNR for the latter index, allowing the
present study to use spectra with SNR-per-pixel as low as 8.
