  We axiomatize the algebraic properties of toroidal compactifications of
(mixed) Shimura varieties and their automorphic vector bundles. A notion of
generalized automorphic sheaf is proposed which includes sheaves of
(meromorphic) sections of automorphic vector bundles with prescribed vanishing
and pole orders along strata in the compactification, and their quotients.
These include, for instance, sheaves of Jacobi forms and weakly holomorphic
modular forms. Using this machinery we give a short and purely algebraic proof
of the proportionality theorem of Hirzebruch and Mumford.
