Algebraic cones of LCK manifolds with potential

Abstract

A complex manifold X, dimX>2, is called “an LCK manifold with potential”, if it can be realized as a complex submanifold of a Hopf manifold. Let X˜ be its Z-covering, considered as a complex submanifold in Cn﹨0. We prove that X˜ is algebraic. We call the manifolds obtained this way the algebraic cones, and show that the affine algebraic structure on X˜ is independent from the choice of X. We give several intrinsic definitions of an algebraic cone, and prove that these definitions are equivalent.