Complex cycles on real algebraic models of a smooth manifold

Abstract

Let M M be a compact connected orientable C ∞ {C^\infty } submanifold of R n {\mathbb {R}^n} with 2 dim ⁡ M + 1 ≤ n 2\dim M + 1 \leq n . Let G G be a subgroup of H 2 ( M , Z ) {H^2}(M,\mathbb {Z}) such that the quotient group H 2 ( M , Z ) {H^2}(M,\mathbb {Z}) has no torsion. Then M M can be approximated in R n {\mathbb {R}^n} by a nonsingular algebraic subset X X such that H C - a l g 2 ( X , Z ) H_{\mathbb {C} \operatorname {- alg}}^{2}(X,\mathbb {Z}) is isomorphic to G G . Here H C - a l g 2 ( X , Z ) H_{\mathbb {C}\operatorname { - alg}}^2(X,\mathbb {Z}) denotes the subgroup of H 2 ( X , Z ) {H^2}(X,\mathbb {Z}) generated by the cohomology classes determined by the complex algebraic hypersurfaces in a complexification of X X .