E8 lattice and the Kodaira dimension of orthogonal modular varieties

TL;DR

This paper proves that orthogonal modular varieties associated with certain lattices become of general type when the lattice is extended by a large enough multiple of the E_8 lattice.

Contribution

It establishes the general type property for orthogonal modular varieties with lattices extended by E_8, advancing understanding of their geometric classification.

Findings

Orthogonal modular varieties are of general type for large lattice extensions.

The result applies to lattices of signature (2,n) extended by E_8.

Provides new insights into the geometry of modular varieties.

Abstract

We prove that for any even lattice L of signature (2,n), the modular variety defined by the orthogonal group of the lattice L+mE_8 is of general type when m is sufficiently large.