Algebraic cones of LCK manifolds with potential

TL;DR

This paper proves that the $ ext{Z}$-covering of LCK manifolds with potential is algebraic and independent of the original manifold, providing intrinsic definitions and equivalences for algebraic cones.

Contribution

It establishes the algebraic nature of the $ ext{Z}$-covering of LCK manifolds with potential and shows the independence of the affine structure from the choice of manifold.

Findings

The $ ext{Z}$-covering is algebraic.

The affine algebraic structure is independent of the original manifold.

Multiple intrinsic definitions of algebraic cones are equivalent.

Abstract

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 $\Z$-covering, considered as a complex submanifold in $C^n \backslash 0$. 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.