A note on locally conformally Kaehler structures and small deformations of Hopf manifolds

2023/05/18 by Hasegawa, Keizo
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2305.10768

Abstract

A Hopf manifold is a compact complex manifold of which the universal covering is Cⁿ\0. In this note we show that any Hopf manifold admits a locally conformally Kaehler structure (shortly lcK structure), by constructing a complex analytic family around a Hopf manifold of diagonal type, which admits a lcK potential, and applying a well known fact (due to Ornea and Verbitsky) that the property of lcK potential is preserved under a complex analytic family over a sufficiently small parameter space.

Related

• LCK metrics on complex spaces with quotient singularities
• Coverings of locally conformally Kähler complex spaces
• Locally conformal Kaehler reduction
• Transformations of compact locally conformally Kähler manifolds
• Indefinite locally conformal Kaehler manifolds