On the splitting of surfaces in motivic stable homotopy category

TL;DR

This paper investigates the conditions under which the motivic stable homotopy type of smooth projective surfaces over perfect fields splits off the top cell, providing new insights and proofs for specific classes like curves and Calabi-Yau surfaces.

Contribution

It offers new criteria for splitting in the motivic stable homotopy category and extends existing results to include K3 and Calabi-Yau surfaces with alternative proofs.

Findings

Criteria for top cell splitting in motivic homotopy types

Alternative proof of curve splitting over fields with char ≠ 2

Understanding Calabi-Yau surface splittings via Chow-Witt correspondences

Abstract

Let $k$ be a perfect field and $X$ be a smooth projective surface over $k$ with a rational point, we discuss the condition of splitting off the top cell for the motivic stable homotopy type of $X$. We also study some outlying examples, such as K3 surfaces. When $k$ is an algebraically field with characteristic not equal to 2, we can give an alternative proof of the splitting result of curves and also understand the splittings of Calabi-Yau surfaces via the motivic Hurewicz theorem and decomposition of the Chow-Witt correspondences.