The Grothendieck-lefschetz Theorem for Normal Projective Varieties

We prove that for a normal projective variety X in characteristic 0, and a base-point free ample line bundle L on it, the restriction map of divisor class groups Cl(X) → Cl(Y ) is an isomorphism for a general member Y ∈ |L| provided that dimX ≥ 4. This is a generalization of the Grothendieck-Lefschetz theorem, for divisor class groups of singular varieties. We work over k, an algebraically closed field of characteristic 0. Let X be a smooth projective variety over k and Y a smooth complete intersection subvariety of X. The Grothendieck-Lefschetz theorem states that if dimension Y ≥ 3, the Picard groups of X and Y are isomorphic. In this paper, we wish to prove an analogous statement for singular varieties, with the Picard group replaced by the divisor class group. Let X be an irreducible projective variety which is regular in codimension 1 (for example, X may be irreducible and normal). Recall that for such X, the divisor class group Cl(X) is defined as the group of linear equivalence classes of Weil divisors on X (see [10], II, §6). If dimX = d, then Cl(X) coincides with the Chow group CHd−1(X) as defined in Fulton’s book [7]. If Y ⊂ X is an irreducible Cartier divisor, which is also regular in codimension 1, there is a well-defined restriction homomorphism