  The $\nu_e-^{56}$Fe cross section is evaluated in the projected quasiparticle
random phase approximation (PQRPA). This model solves the puzzle observed in
RPA for nuclei with mass around $^{12}$C, because it is the only RPA model that
treats the Pauli principle correctly. The cross sections as a function of the
incident neutrino energy are compared with recent theoretical calculations of
similar models. The average cross section weighted with the flux spectrum
yields a good agreement with the experimental data. The expected number of
events in the detection of supernova neutrinos is calculated for the LVD
detector leading to an upper limit for the electron neutrino energy of
particular importance in this experiment
