Lefschetz theorems for torsion algebraic cycles in codimension 2

Abstract

Let Y be a smooth projective variety over C, and X be a smooth hypersurface in Y. We prove that the natural restriction map on Chow groups of codimension two cycles is an isomorphism when restricted to the torsion subgroups provided dim⁡Y≥5. We prove an analogous statement for a very general hypersurface X⊂P4 of degree ≥5. In the more general setting of a very general hypersurface X of sufficiently high degree in a fixed smooth projective four-fold Y, under some additional hypothesis, we prove that the restriction map is an isomorphism on ℓ-primary torsion for almost all primes ℓ. As a consequence, we obtain a weak Lefschetz theorem for torsion in the Griffiths groups of codimension 2 cycles, and prove the injectivity of the Abel–Jacobi map when restricted to torsion in this Griffiths group, thereby providing a partial answer to a question of Nori.