Toroidal compactifications of PEL-type Kuga families

We explain how compactifications of Kuga families of abelian varieties over PEL-type Shimura varieties, including for example all those products of universal abelian schemes, can be constructed (up to good isogenies not affecting the relative cohomology) by a uniform method. We also calculate the relative cohomology and explain its various properties crucial for applications to the cohomology of automorphic bundles.