On the smooth compactification of Siegel spaces (Q1312151)

For every bounded Hermitian symmetric domain \(\Omega\) and every near arithmetic subgroup \(\Gamma\) of \(\Aut \Omega\) the quotient \(X: = \Gamma \backslash \Omega\) is a quasiprojective manifold and admits a smooth (minimal) toroidal compactification \(\overline X\) [see \textit{A. Ash}, \textit{D. Mumford}, \textit{M. Rapoport} and \textit{Y. Tai}, `Smooth compactification of locally symmetric varieties' (1975; Zbl 0334.14007) and \textit{Y. Namikawa}, `Toroidal compactification of Siegel spaces' (1980; Zbl 0466.14011)]. The author studies the structure of the boundary divisor \(D: = \overline X \backslash X\) and the canonical bundle \(K_{\overline X}\) of \(\overline X\) for \(\Omega: = S_ n\), the Siegel upper half space of rang \(n\), and \(\Gamma: = \Gamma (k)\), \(k \geq 3\), a (neat) principal congruence subgroup of \(Sp (n,\mathbb{Z})\). He gets precise and detailed results in the case \(n=2\), in particular for the canonical bundle \(K_{\overline X}\) (Theorem 3.1) and for the description of the singular canonical volume form of \(\overline X\) along \(D\) (Theorem 4.1). Furthermore he shows that \(\Gamma (k) \backslash S_ 2\) is of general type for \(k \geq 4\) (Theorem 3.2).