Motivic multiplicativity of complete intersections

Abstract

For a smooth projective variety equipped with a Chow-K\"unneth (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.