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Algebraic cycles and special Horikawa surfaces. (English) Zbl 1477.14011

Summary: This note is about a certain 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 G. Pearlstein and Z. Zhang [Algebr. Geom. 6, No. 2, 132–147 (2019; Zbl 1427.14071)] and by A. Garbagnati [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 19, No. 1, 345–386 (2019; Zbl 1419.14059)], are related to \(K3\) surfaces. We show that special Horikawa surfaces have a multiplicative Chow-Künneth decomposition, in the sense of Shen-Vial. As a consequence, the Chow ring of special Horikawa surfaces displays \(K3\)-like behavior.

MSC:

14C15 (Equivariant) Chow groups and rings; motives
14C25 Algebraic cycles
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
14J28 \(K3\) surfaces and Enriques surfaces
14J29 Surfaces of general type
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