  We measure the evolution of the galaxy stellar mass function from z=1.3 to
z=0.5 using the first 53,608 redshifts of the ongoing VIMOS Public
Extragalactic Survey (VIPERS). We estimate the galaxy stellar mass function at
several epochs discussing in detail the amount of cosmic variance affecting our
estimate. We find that Poisson noise and cosmic variance of the galaxy mass
function in the VIPERS survey are comparable with the statistical uncertainties
of large surveys in the local universe. VIPERS data allow us to determine with
unprecedented accuracy the high-mass tail of the galaxy stellar mass function,
which includes a significant number of galaxies that are usually too rare to
detect with any of the past spectroscopic surveys. At the epochs sampled by
VIPERS, massive galaxies had already assembled most of their stellar mass. We
apply a photometric classification in the (U-V) rest-frame colour to compute
the mass function of blue and red galaxies, finding evidence for the evolution
of their contribution to the total number density budget: the transition mass
above which red galaxies dominate is found to be about 10^10.4 M_sun at z=0.55
and evolves proportionally to (1+z)^3. We are able to trace separately the
evolution of the number density of blue and red galaxies with masses above
10^11.4 M_sun, in a mass range barely studied in previous work. We find that
for such large masses, red galaxies show a milder evolution with redshift, when
compared to objects at lower masses. At the same time, we detect a population
of similarly massive blue galaxies, which are no longer detectable below z=0.7.
These results show the improved statistical power of VIPERS data, and give
initial promising indications of mass-dependent quenching of galaxies at z~1.
[Abridged]
