Let \(A_ n=H_ n/\Gamma_ n\), where \(H_ n\) is the Siegel space of degree n and \(\Gamma_ n=Sp_{2n}({\mathbb{Z}})\). \(A_ n\) is called a Siegel modular variety. The main result of this paper is to prove Theorem 2. Let \(n\geq 10\). Then any subvariety in \(A_ n\) of codimension one is of general type. This is a solution to a conjecture raised by E. Freitag [Prog. Math. 46, 93-113 (1984; Zbl 0546.10024)].
The proof is sketched as follows. First the author constructs a matricial multi-tensor of differential forms by using theta-series with spherical functions defined on \(H_ n\). Then he pursues the proof of his theorem with the help of the ideas in Freitag’s cited work and the work of Y.-S. Tai [Invent. Math. 68, 425-439 (1982; Zbl 0508.14038)].