Algebraic cones of LCK manifolds with potential

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 $\mathbb{Z}$-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.

28 pages, LaTeX, v. 2.0, minor error corrected, intro rewritten

References in corpus (2)

• Normal form of bimeromorphically contractible holomorphic Lagrangian submanifolds
• Non-linear Hopf manifolds are locally conformally Kahler

Cited by in corpus (1)

• Holomorphic tensors on Vaisman manifolds