On toric locally conformally Kähler manifolds

Abstract

We study compact toric strict locally conformally Kähler manifolds. We show that the Kodaira dimension of the underlying complex manifold is $-\infty$ and that the only compact complex surfaces admitting toric strict locally conformally Kähler metrics are the diagonal Hopf surfaces. We also show that every toric Vaisman manifold has lcK rank 1 and is isomorphic to the mapping torus of an automorphism of a toric compact Sasakian manifold.

17 pages

References in corpus (3)

• LCK rank of locally conformally Kahler manifolds with potential
• Toric Vaisman Manifolds
• A characterisation of toric LCK manifolds

Cited by in corpus (8)

• A characterisation of toric LCK manifolds
• Locally conformally Kähler manifolds with holomorphic Lee field
• Locally conformally symplectic convexity
• LcK structures with holomorphic Lee vector field on Vaisman-type manifolds
• Locally conformally Kähler solvmanifolds: a survey
• LCK metrics on toric LCS manifolds
• Existence Criteria for LCK Metrics
• Closed 1-Forms and Twisted Cohomology