On the existence of holomorphic embeddings of strictly pseudoconvex algebraic hypersurfaces into spheres

We show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into a sphere in $\bC^{N+1}$ for any $N$. In fact, we show that there are strictly pseudoconvex, real algebraic hypersurfaces in $\bC^{n+1}$ that cannot be locally embedded into any compact, strictly pseudoconvex, real algebraic hypersurface.