  An extended database of experimental data is needed to address uncertainties
of the nuclear-physics input parameters for Hauser-Feshbach calculations.
Especially $\alpha$+nucleus optical model potentials at low energies are not
well known. The in-beam technique with an array of high-purity germanium (HPGe)
detectors was successfully applied to the measurement of absolute cross
sections of an ($\alpha$,$\gamma$) reaction on a heavy nucleus at sub-Coulomb
energies. The total and partial cross-section values were measured by means of
in-beam $\gamma$-ray spectroscopy. Total and partial cross sections were
measured at four different $\alpha$-particle energies from $E_\alpha = 10.5$
MeV to $E_\alpha = 12$ MeV. The measured total cross-section values are in
excellent agreement with previous results obtained with the activation
technique, which proves the validity of the applied method. The experimental
data was compared to Hauser-Feshbach calculations using the nuclear reaction
code TALYS. A modified version of the semi-microscopic $\alpha$+nucleus optical
model potential OMP 3, as well as modified proton and $\gamma$ widths, are
needed in order to obtain a good agreement between experimental data and
theory. It is found, that a model using a local modification of the
nuclear-physics input parameters simultaneously reproduces total cross sections
of the $^{112}$Sn($\alpha$,$\gamma$) and $^{112}$Sn($\alpha$,p) reactions. The
measurement of partial cross sections turns out to be very important in this
case in order to apply the correct $\gamma$-ray strength function in the
Hauser-Feshbach calculations. The model also reproduces cross-section values of
$\alpha$-induced reactions on $^{106}$Cd, as well as of ($\alpha$,n) reactions
on $^{115,116}$Sn, hinting at a more global character of the obtained
nuclear-physics input.
