  We use a combined generalized spectroscopic ellipsometry and density
functional theory approach to determine and analyze the anisotropic dielectric
functions of an $\alpha$-Ga$_2$O$_3$ thin film. The sample is grown epitaxially
by plasma-assisted molecular beam epitaxy on $m$-plane sapphire. Generalized
spectroscopic ellipsometry data from multiple sample azimuths in the spectral
range from 0.73 eV to 8.75 eV are simultaneously analyzed. Density functional
theory is used to calculate the valence and conduction band structure. We
identify, for the indirect-bandgap material, two direct band-to-band
transitions with $M_0$-type van Hove singularities for polarization
perpendicular to the $c$ axis, $E_{0,\perp}=5.46(6)$ eV and
$E_{0,\perp}=6.04(1)$ eV, and one direct band-to-band transition with
$M_1$-type van Hove singularity for polarization parallel with
$E_{0,||}=5.44(2)$ eV. We further identify excitonic contributions with small
binding energy of 7 meV associated with the lowest ordinary transition, and a
hyperbolic exciton at the $M_1$-type critical point with large binding energy
of 178 meV.
