Algebraic cycles and triple $K3$ burgers

Abstract

We consider surfaces of geometric genus 3 with the property that their transcendental cohomology splits into 3 pieces, each piece coming from a K3 surface. For certain families of surfaces with this property, we can show there is a similar splitting on the level of Chow groups (and Chow motives).

Triple Covers of K3 Surfaces

“…The surfaces with such that the transcendental Hodge structure splits in the direct sum of two K3-type Hodge structures are studied in several context (see, e.g., [G3], [L1], [L2], [Mo], [PZ]), and in general, it is interesting to look for K3 surfaces geometrically associated with the K3-type Hodge structure. In the case of covers of K3 surfaces S (and in particular in the setting of Corollary 4.3), at least one of the two K3-type Hodge structures is of course geometrically associated with the K3 surface S ; indeed, it is the pull back of the Hodge structure of S .…”

“…The Galois covers of K3 surfaces are a quite classical and interesting argument of research: for example, the K3 surfaces which are Galois covers of other K3 surfaces are classified in [X], and the abelian surfaces which are Galois covers of K3 surfaces are classified in [Fu]. The study of surfaces with higher Kodaira dimension which are covers of K3 surfaces is less systematic, and sporadic examples appear in order to construct specific surfaces (see, e.g., [CD], [L1], [L2], [PZ], [RRS], [Sa]. Nevertheless, a systematic approach to the study of the double covers of K3 surfaces is presented in [G3], where smooth double covers are classified and their birational invariants are given.…”

“…Nevertheless, a systematic approach to the study of the double covers of K3 surfaces is presented in [G3], where smooth double covers are classified and their birational invariants are given. In the same paper, certain cover surfaces with are described with more details, since the geometry of these surfaces is quite interesting (see, e.g., [L1], [L2]).…”