Complex cycles on real algebraic models of a smooth manifold

Abstract:

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