On the Kodaira dimensions of Hilbert modular varieties
Let K be a totally real algebraic number field of degree \(n>1\). The group \(\Gamma =SL_ 2({\mathfrak O}_ K)\) acts on \({\mathbb{H}}^ n\) by the modular substitution. For a subgroup G of Aut(K/\({\mathbb{Q}})\), acting on \({\mathbb{H}}^ n\) as permutations of the coordinates, denote by \({\hat \Gamma}\) the composite of G and \(\Gamma\). The variety \(X:={\mathbb{H}}^ n/{\hat \Gamma}\) can be compactified by adding a finite number of cusps. The non-singular model \(\tilde X\) of this compactification is called Hilbert modular variety if \(G=\{1\}\) and symmetric Hilbert modular variety if \(G\neq \{1\}.\) The author shows that \(\tilde X\) is of general type except for a finite number of fields K. In particular all Hilbert modular varieties of dimension larger than six and all symmetric Hilbert modular varieties of dimension larger than nine are of general type. - Denote by \(X_ 0\) the complement of the fixed points in X. To each (symmetric) Hilbert modular form of weight k corresponds a section of \(H^ 0(X_ 0,\Omega^{\otimes k})\). In his proof the author uses a criterion of Tai to show that for \(n>2\), except for a finite number of fields K, these sections extend to all of X. Then he discusses their extendability to \(\tilde X\) and shows, that for large N, the dimension of \(H^ 0(\tilde X,\Omega^{\otimes N})\) growths like \(N^ n\), in other words, the Kodaira dimension is equal to n and \(\tilde X\) is of general type. In the case of \(n=2\), the author refers to results of \textit{F. Hirzebruch} and \textit{D. Zagier} [in Complex Anal. algebr. Geom., Collect. Pap. dedic. K. Kodaira, 43-77 (1977; Zbl 0354.14011)] in the case of Hilbert modular varieties and to the thesis of \textit{D. Bassendowski} [''Klassifikation Hilbertscher Modulflächen'', Bonn, Math. Schr. 163 (1985)] in the case of symmetric Hilbert modular varieties.
symmetric Hilbert modular variety of general
scientific article; zbMATH DE number 3920663
scientific article