Chow groups of smooth varieties fibred by quadrics

2012/03/12 by Charles Vial, Vial, Charles
Mathematics · #14C15 #14C25 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1203.2651

openalex publication_date 2012/03/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let f : X → B be a proper flat dominant morphism between two smooth quasi-projective complex varieties X and B. Assume that there exists an integer l such that all closed fibres X_b of f satisfy CH_j(X_b) = \Q for all j ≤ l. Then we prove an analogue of the projective bundle formula for CH_i(X) for i ≤ l. When B is a surface, X is projective and l = \lfloor (dim X - 3)/(2) \rfloor, this makes it possible to construct a Chow-Künneth decomposition for X that satisfies Murre's conjectures. For instance we prove Murre's conjectures for complex smooth projective varieties X fibred over a surface (via a flat morphism) by quadrics, or by complete intersections of dimension 4 of bidegree (2,2).

Citations

• On a conjectural filtration on the Chow groups of an algebraic variety

Related

• Murre's conjectures for certain product varieties
• Remarks on Murre's conjecture on Chow groups
• Chow--Kuenneth decomposition for special varieties
• Chow groups of intersections of quadrics via homological projective duality and (Jacobians of) noncommutative motives
• Motives of quadric bundles