Generators of the algebraic symplectic bordism ring

TL;DR

This paper investigates the structure of the algebraic symplectic bordism ring using motivic spectra, spectral sequences, and refined geometric constructions to identify generators and understand its algebraic properties.

Contribution

It introduces a refined Pontryagin-Thom construction and analyzes the motivic Adams spectral sequence to describe the symplectic bordism ring and its generators.

Findings

Description of the $( ext{MSp}^ ext{wedge}_ ext{eta})^*$ ring structure

Identification of symplectic bordism classes from varieties with a 'symplectic twist'

A criterion for selecting generators of the symplectic bordism ring

Abstract

In this paper, we study the $\eta$-completed part of the motivic spectrum $\text{MSp}$ constructed by Panin and Walter, representing the universal $\text{Sp}$-oriented cohomology theory. In particular, we investigate the inclusion $(\text{MSp}^\wedge_\eta)^*\hookrightarrow \text{MGL}^*$ of the cofficient rings, by studying the motivic Adams spectral sequence associated to $\text{MSp}$, mimiking a strategy used by Levine,Yang, Zhao for $\text{MSL}^*$. In order to give a description of $(\text{MSp}^\wedge_\eta)^*$, we refine the Pontryagin-Thom construction in a way that allows one to obtain symplectic bordism classes from a large family of varieties that carry a certain "symplectic twist", and we prove a criterion to select generators among these classes.