  We prove that the ring of Siegel modular forms of weight divisible by g+n+1
is isomorphic to the ring of (log) pluricanonical forms on the n-fold Kuga
family of abelian varieties and its certain compactifications, for every
arithmetic group for a symplectic form of rank 2g>2. We also give applications
to the Kodaira dimension of the Kuga variety. In most cases, the Kuga variety
has canonical singularities.
