Geometric Theory of Parshin's Residues. Toric Neighborhoods of Parshin's Points

TL;DR

This paper develops a geometric framework for understanding Parshin's residues using toric neighborhoods, lattice flags, and singularity resolution techniques, providing a new coordinate system for Parshin's points.

Contribution

It introduces a novel geometric approach to Parshin's residues through lattice flags and toric varieties, linking local algebraic structures with geometric resolutions.

Findings

Established a connection between lattice flags and toric neighborhoods.

Analyzed the local geometry near Parshin's points using singularity resolution.

Provided a coordinate system for Parshin's points based on toric varieties.

Abstract

The paper consist of two parts. In the first part we introduce flags of lattices and associated injective systems of (non-normal) cones and projective systems of (non-normal) affine toric varieties. We study the associated field of multidimensional Laurent power series. In the second part we use the resolution of singularities techniques to study the geometry near a complete flag of subvarieties and the Parshin's residues. The first part plays the role of a standard coordinate neighborhood for Parshin's points.