×

Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type. (English) Zbl 1142.14023

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)

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
PDFBibTeX XMLCite
Full Text: arXiv EuDML EMIS