Algebraic cycles and special Horikawa surfaces

TL;DR

This paper studies a 16-dimensional family of special Horikawa surfaces, showing they have a multiplicative Chow-K"unneth decomposition and exhibit K3-like properties in their Chow ring.

Contribution

It proves that special Horikawa surfaces possess a multiplicative Chow-K"unneth decomposition, linking their Chow ring structure to that of K3 surfaces.

Findings

Special Horikawa surfaces have a multiplicative Chow-K"unneth decomposition.

Their Chow ring exhibits K3-like behaviour.

The results connect these surfaces to broader K3 surface properties.

Abstract

This note is about a $16$-dimensional family of surfaces of general type with $p_g=2$ and $q=0$ and $K^2=1$, called "special Horikawa surfaces". These surfaces, studied by Pearlstein-Zhang and by Garbagnati, are related to K3 surfaces. We show that special Horikawa surfaces have a multiplicative Chow-K\"unneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of special Horikawa surfaces displays K3-like behaviour.