On the splitting of surfaces in motivic stable homotopy category

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.