Algebraic cycles and intersections of three quadrics

Abstract

Let Y be a smooth complete intersection of three quadrics, and assume the dimension of Y is even. We show that Y has a multiplicative Chow–Künneth decomposition, in the sense of Shen–Vial. As a consequence, the Chow ring of (powers of) Y displays K3-like behaviour. As a by-product of the argument, we also establish a multiplicative Chow–Künneth decomposition for double planes.

Some more Fano threefolds with a multiplicative Chow-Künneth decomposition

“…For more background on the concept of MCK, and for examples of varieties with an MCK decomposition, we refer to [47,Section 8], as well as [53], [48], [13], [28], [39], [29], [30], [31], [12], [33], [34], [36], [42].…”

Algebraic cycles and Fano threefolds of genus 8

“…The property of having an MCK decomposition is motivated by, and closely related to, Beauville's "splitting property' conjecture" [2]. [58], [51], [10], [28], [38], [29], [30], [31], [9], [33], [34], [36].…”

Some Motivic Properties of Gushel-Mukai Sixfolds

“…In dimension two, a smooth quartic in P 3 has an MCK decomposition, but a very general surface of degree ≥ 7 in P 3 should not have one [15,Proposition 3.4]. For a discussion in greater detail, and further examples of varieties with an MCK decomposition, the reader may check [51, Section 8], as well as [55], [52], [16], [29], [30], [31], [32], [33], [34], [35], [36], [37], [15], [38], [39], [40] We say that Y → B has the Franchetta property if Y → B has the Franchetta property in codimension j for all j.…”