Motivic multiplicativity of complete intersections
Ze Xu
TL;DR
This work develops a comprehensive framework for understanding how Chow–Künneth (CK) decompositions interact with intersection products on smooth projective varieties. It introduces motivic m-fold twist-multiplicativity and multiplicativity defects, defined via the decomposition of the $(m+1)$-st small diagonal, and then establishes criteria and stability results under standard geometric operations. The authors apply this framework to curves, surfaces, and ample subvarieties with trivial Chow groups, and derive explicit formulas and bounds for the motivic $2$-fold defect of Fano and Calabi–Yau complete intersections in weighted projective spaces, proving $0$-multiplicativity in many CY/Fano hypersurfaces. They further develop isogenous correspondences and cycle-relations to control obstructions, and obtain significant applications such as Franchetta properties for universal families and multiplicative decomposition phenomena in derived categories. The results advance the understanding of the motivic structure of complete intersections and provide evidence supporting Voisin’s conjectures in the CY setting, with broad implications for the study of Chow motives and diagonal cycles.
Abstract
For a smooth projective variety equipped with a Chow-Künneth (abbr. CK) decomposition, the notions of motivic multiple twist-multiplicativity and multiplicativity defect are introduced to interpret the obstruction to the compatibility of the multiple intersection product with its CK decompositions, generalizing the more restrictive notion of multiplicativity introduced in [31]. We establish the basic properties of these notions. Then we show that the multiplicativity defects of curves, surfaces and ample subvarieties in varieties with trivial Chow groups have reasonable upper bounds. Furthermore, we determine explicitly the motivic 2-fold multiplicativity defect for any Fano or Calabi-Yau complete intersection in a smooth weighted projective space, strengthening a main result of [11] in the Calabi-Yau case. Particularly, any Fano or Calabi-Yau hypersurface admits motivic 0-multiplicativity, generalizing the case of cubic hypersurfaces proved in [10] and [13], and conforming a conjecture of Voisin [35] in the Calabi-Yau case. As a consequence, certain relative powers of the corresponding universal families satisfy the Franchetta property. We also provide several other applications.