Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type

TL;DR

This paper investigates the Kodaira dimension of orthogonal type moduli spaces, establishing that certain modular varieties are of general type for sufficiently large parameters, thus advancing understanding of their geometric properties.

Contribution

It provides the first results on the Kodaira dimension of high-dimensional orthogonal type moduli spaces, specifically proving they are of general type when parameters meet certain conditions.

Findings

Modular variety of signature (2, 8m+2) is of general type for m ≥ 5.

First results on Kodaira dimension for dimensions >19.

Advances understanding of the geometry of orthogonal type moduli spaces.

Abstract

For many classical moduli spaces of orthogonal type there are results about the Kodaira dimension. But nothing is known in the case of dimension greater than 19. In this paper we obtain the first results in this direction. In particular the modular variety defined by the orthogonal group of the even unimodular lattice of signature (2, 8m+2) is of general type if m is at least 5.