Motivic multiplicativity of complete intersections

Abstract

For a smooth projective variety endowed with a Chow-K\"unneth (abbr. CK) decomposition, we introduce the notions of motivic multiple twist-multiplicativity and multiplicativity defect to measure the obstruction to the compatibility of multiple intersection products with the given CK decomposition. These notions extend the more restrictive notion of multiplicativity introduced by Shen-Vial. We establish their basic properties and derive natural upper bounds for the motivic multiplicativity defects of curves, surfaces, and ample subvarieties of varieties with trivial Chow groups. We then explicitly determine the motivic 2-fold multiplicativity defect of any smooth Fano or Calabi-Yau complete intersection in a smooth weighted projective space, thereby strengthening a result of Fu in the Calabi-Yau case. In particular, we prove that any smooth Fano or Calabi-Yau hypersurface admits motivic 0-multiplicativity. This generalizes the corresponding result for cubic hypersurfaces, proved independently by Diaz and Fu-Laterveer-Vial, and confirms a conjecture of Voisin in the Calabi-Yau case. As a consequence, certain relative powers of the associated universal families satisfy the Franchetta property. We also obtain several further applications.