Borcherds products approximating Gersten complex

TL;DR

This paper constructs a complex related to orthogonal modular varieties using Borcherds products and Milnor K-theory, offering a new approach to studying higher Chow groups via special cycles.

Contribution

It introduces a novel complex that links Borcherds products with the Gersten complex, facilitating the study of higher Chow groups of modular varieties.

Findings

Established a formalism connecting Borcherds products to higher Chow groups.

Constructed a complex resembling the Gersten complex in Milnor K-theory.

Provided a new framework for analyzing special cycles on modular varieties.

Abstract

For an orthogonal modular variety, we construct a complex which is defined in terms of lattices and elliptic modular forms, which resembles the Gersten complex in Milnor K-theory, and which has a morphism to the Gersten complex of the modular variety by the Borcherds lifting. This provides a formalism for approaching the higher Chow groups of the modular variety by special cycles and Borcherds products. The construction is an incorporation of the theory of Borcherds products and ideas from Milnor K-theory.