Algebraic cones of LCK manifolds with potential

A complex manifold $X$ is called “LCK manifolds with potential” if it can be realized as a complex submanifold of a Hopf manifold. Let $Y$ its $\mathbb{Z}$-covering, considered as a complex submanifold in $C^n\mathrm{\backslash }0$. We prove that $Y$ is algebraic. We call the manifolds obtained this way the algebraic cones, and show that the affine algebraic structure on $Y$ is independent from the choice of $X$. We give several intrinsic definitions of an algebraic cone, and prove that these definitions are equivalent.