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.

中文翻译：

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广义自同构滑轮与HIRZEBRUCH-MUMFORD比例原理

我们公理化（混合）志村簇及其自同构向量丛的环形紧化的代数性质。提出了广义自同构束的概念，其中包括自同构向量丛的（亚同质）部分的束，其在紧致化中沿层具有规定的消失阶和极阶，以及它们的商。例如，这些包括雅可比形式的滑轮和弱全纯模形式。使用这个机制，我们给出了 Hirzebruch 和 Mumford 比例定理的简短且纯代数的证明。