Torsion in the Chow group of codimension two: The case of varieties with isolated singularities
Let X be a nonsingular quasiprojective algebraic variety over an algebraically closed field k, let \(CH^ 2(X)\) be the group of codimension 2 cycles modulo rational equivalence, and let \({}_ nCH^ 2(X)\) be the kernel of multiplication by n in \(CH^ 2(X)\). From a result of \textit{A. S. Merkur'ev} and \textit{A. A. Suslin} on \(K_ 2\) [cf. Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)] it follows that if n is prime to char k, then \({}_ nCH^ 2(X)\) is finite. The author extends this result to (a suitably defined) Chow group of algebraic varieties with isolated singularities. As an application, the author shows that if Y is a complete surface with isolated singularities, there is an isomorphism \(CH^ 2(Y)(\ell){\tilde \to}Alb(Y')(\ell)\), where Y' is the desingularization of Y and \(CH^ 2(Y)(\ell)\) is the subgroup of elements whose order is a power of \(\ell\).
codimension 2 cycles modulo rational equivalence
Chow group of algebraic varieties with isolated singularities
scientific article; zbMATH DE number 3946313
scientific article