Gritsenko, V.; Hulek, K.; Sankaran, G. K. Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type. (English) Zbl 1142.14023 Doc. Math. 13, 1-19 (2008). Given an integral indefinite lattice \(L\) of signature \((2,n)\) with the associated bilinear form \(f\), let \[ D_L:= \{[w]\mid[w]\in \mathbb{P}(L\otimes \mathbb{C}),\,f(w,w)= 0,\, f(w,\overline w)> 0\}^+ \] be a connected component of the homogeneous type IV complex domain of dimension \(n\) and let \({\mathcal F}_L(\Gamma:= \Gamma\setminus D_L\), where \(\Gamma\)’ is a finite index subgroup of \(O^+(L)\), the integral orthogonal group leaving \(D_L\) invariant. Recently the authors [Doc. Math., J. DMV 12, 215–241 (2007; Zbl 1127.11038)] have obtained an exact formula for the Hirzebruch-Mumford volume of the orthogonal group. In another work [Invent. Math., 169, No. 3, 519–567 (2007; Zbl 1128.14027)], the authors prove existence of a good toroidal compactification of the modular variety \({\mathcal F}_L(\Gamma)\). In the reviewed paper, making use of their cited results, the authors determine the Kodaira dimension of some quasi-projective varieties for the two series of even lattices \(2U\oplus mE_8(-1)\) and \(2U\oplus mE_8(-1)\oplus\langle-2d\rangle\), where \(U\) is the hyperbolic plane, \(E_8(-1)\) is the negative definite lattice associated with the root system \(E_8\), \(m\in\mathbb{N}\), and \(\langle-2d\rangle\) stands for the lattice generated by a vector of square \(-2d\). Infinitely many of the varieties studied in the paper turn out to be of general type. Reviewer: B. Z. Moroz (Bonn) Cited in 8 Documents MSC: 14J15 Moduli, classification: analytic theory; relations with modular forms 11F55 Other groups and their modular and automorphic forms (several variables) 14M17 Homogeneous spaces and generalizations 14J28 \(K3\) surfaces and Enriques surfaces 14J29 Surfaces of general type Keywords:locally symmetric spaces; quadratic forms; Kodaira dimension; orthogonal groups; Hirzebruch-Mumford proportionality Citations:Zbl 1127.11038; Zbl 1128.14027 PDFBibTeX XMLCite \textit{V. Gritsenko} et al., Doc. Math. 13, 1--19 (2008; Zbl 1142.14023) Full Text: arXiv EuDML EMIS