We axiomatise the algebraic properties of toroidal compactifications of (mixed) Shimura varieties and their automorphic vector bundles. A notion of generalised 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.

In this paper, we show that the Chern classes ck of the de Rham bundle ${\cal H}_{{\rm d}R}$ defined on any ‘good’ toroidal compactification $\bar{\cal A}_g$ of the moduli space of Abelian varieties of dimension g are zero in the rational Chow ring of $\bar{\cal A}_g$, for g = 4, 5 and k>0.

Let f:X→S be a projective morphism of Noetherian schemes. We assume f purely of relative dimension d and finite Tor-dimensional. We associate to d+1 invertible sheaves $\cal L$1,$\ldots$,$\cal L$d+1 on X a line bundle IX/S($\cal L$1,$\ldots$,$\Cal L$d+1) on S depending additively on the $\cal L$i, commuting to ‘good’ base changes and which represents the integral along the fibres of f of the product of the first Chern classes of the $\cal L$i. If d=0, IX/S($\cal L$) is the norm $\cal N$X/S($\cal L$).