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 certain compactifications of it, 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.

中文翻译：

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通用阿贝尔簇和 Siegel 模形式

我们证明了重量的 Siegel 模形式的环可以被整除 $G+n+1$ 同构于 (log) pluricanonical 形式的环 $n$-fold 阿贝尔变体的 Kuga 族和它的某些紧化，对于每一个算术群的秩的辛形式 $2G>2$. 我们还应用了 Kuga 品种的 Kodaira 维度。在大多数情况下，库加变种具有规范奇点。