  Atomically thin films of III-VI post-transition metal chalcogenides (InSe and
GaSe) form an interesting class of two-dimensional semiconductor that feature
strong variations of their band gap as a function of the number of layers in
the crystal [1-4] and, specifically for InSe, an earlier predicted crossover
from a direct gap in the bulk [5,6] to a weakly indirect band gap in monolayers
and bilayers [7-11]. Here, we apply angle resolved photoemission spectroscopy
with submicrometer spatial resolution ($\mu$ARPES) to visualise the
layer-dependent valence band structure of mechanically exfoliated crystals of
InSe. We show that for 1 layer and 2 layer InSe the valence band maxima are
away from the $\mathbf{\Gamma}$-point, forming an indirect gap, with the
conduction band edge known to be at the $\mathbf{\Gamma}$-point. In contrast,
for six or more layers the bandgap becomes direct, in good agreement with
theoretical predictions. The high-quality monolayer and bilayer samples enables
us to resolve, in the photoluminescence spectra, the band-edge exciton (A) from
the exciton (B) involving holes in a pair of deeper valence bands, degenerate
at $\mathbf{\Gamma}$, with the splitting that agrees with both $\mu$ARPES data
and the results of DFT modelling. Due to the difference in symmetry between
these two valence bands, light emitted by the A-exciton should be predominantly
polarised perpendicular to the plane of the two-dimensional crystal, which we
have verified for few-layer InSe crystals.
