  The article has two parts. The first part is devoted to proving a singular
version of the logarithmic Kodaira-Akizuki-Nakano vanishing theorem of Esnault
and Viehweg. This is then used to prove other vanishing theorems. In the second
part these vanishing theorems are used to prove an Arakelov-Parshin type
boundedness result for families of canonically polarized varieties with
rational Gorenstein singularities.
