Bihermitian metrics on Hopf surfaces

TL;DR

This paper constructs strongly bihermitian metrics on specific Hopf surfaces, extending previous locally conformally Kaehler metrics, and classifies which compact complex surfaces admit such metrics.

Contribution

It introduces new strongly bihermitian metrics on Hopf surfaces and completes the classification of compact complex surfaces with these metrics.

Findings

Constructed strongly bihermitian metrics on certain Hopf surfaces

Proved non-existence of such metrics on Inoue surfaces with zero second Betti number

Extended the classification of complex surfaces admitting bihermitian metrics

Abstract

Inspired by a construction due to Hitchin, we produce strongly bihermitian metrics on certain Hopf complex surfaces, which integrate the locally conformally Kaehler metrics found by Gauduchon and Ornea. We also show that the Inoue complex surfaces with zero second Betti number do not admit bihermitian metrics. This completes the classification of the compact complex surfaces admitting strongly bihermitian metrics.